## Geometry Chapter 1 Angle Triangle And Quadrilateral Exercise 1 Solved Example Problems

**Angle Triangle and Quadrilater al **

**Introduction**In order to verify different axioms in geometry and to construct various geometrical figures, the knowledge of angles, triangles, and quadrilaterals is essential.

So, in this chapter, our aim is to discuss different types of angles, triangles, and quadrilaterals. The discussions of this chapter will help the students a lot in the future.

**Wbbse Class 7 Maths Solutions**

**Angle**

When two line segments meet at a point, an angle is formed. Those two line segments are called the arms of that angle and the point is called the vertex of the angle.

**Read and Learn More WBBSE Solutions For Class 7 Maths**

The line segments AB and AC have met at a point A and the angle ∠BAC is formed.

AB and AC are the two arms of the angle and A is the vertex.

If we assume a point D on AS and another point E on AC then ∠DAE and ∠BAC will be of the same measure.

**Adjacent angles**

If two angles have the same vertex and one common arm and if the two angles are on opposite sides of the common arm then the two angles are called adjacent angles.

∠POQ and ∠QOR are adjacent angles because the vertex of both the angles is O and their common arm is OQ and the two angles are on opposite sides of this common arm.

**Perpendicular and Right angle**

If a straight line stands on another straight line in such a way.that, two adjacent angles are equal then one of the straight lines is called a perpendicular to the other. Each of the two adjacent angles is called a right angle.

The straight line OC is perpendicular on AB.

Both ∠AOC and ∠BOC are right angles.

1 right angle = 90°.

Here point 0 is the foot of the perpendicular drawn from C on AB.

**Straight angle**

A straight line AB is drawn on a paper. If point C is taken on the straight line AB then ∠ACB will be a straight angle. 1 straight angle = 180° = 2 right angles.

**Wbbse Class 7 Maths Solutions**

**Acute angle, Obtuse angle, and Reflex angle**

**Acute angle:** An angle that is less than a right angle is called an acute angle.

For example, 30°, 44°, 70°, etc., are acute angles. Since in the figure, 0°<x<90^{o}, hence ∠ABC = ∠x is an acute angle.

**Obtuse angle:** An angle that is greater than one right angle but less than two right angles is called an obtuse angle.

For example, 95°, 110°, 145°, etc., are obtuse angles. Since in the figure 90°<y<180°, hence ∠DEF = ∠y is an obtuse angle.

**Reflex angle:** An angle which is greater than two right angles but less than four right angles is called a reflex angle.

For example 190°, 210°, 300°, etc., are reflex angles. Since in the figures, 180°<z<360°, hence ∠AOB =∠Z is reflex angle in both the images.

**Complementary angle and Supplementary angle**

**Complementary angle:** If the sum of two angles is equal to 90° or one right angle then each angle is called the complementary angle to the other angle.

In the image, ∠BOC is the complementary angle to ∠AOC. In this case, x + y = 90°

**Example**** :** 20° is complementary to 70° as (90°-20°) = 70°.

**Supplementary angle:** If the sum of two angles is equal to 180° or two right angles then each angle is called the supplementary angle to the other angle. In the image, ∠BOC is the supplementary angle to ∠AOC. In the case, x + y = 180°.

**Example**** : **100° is supplementary to 80° as (180°-100°) = 80°.

**Wbbse Class 7 Maths Solutions**

**Vertically opposite angles**

If two straight lines intersect each other, two pairs of angles are formed on the opposite sides of the intersecting point.

Then any angle of a pair of angles is called vertically opposite angle of the other.

The straight lines AB and CD intersect at the point O. ∠BOD is the vertically opposite angle of ∠AOC and ∠AOD is the vertically opposite angle of ∠BOC.

The vertically opposite angles are always of the same measure.

∴ ∠AOC = ∠BOD and ∠BOC = ∠AOD.

**Transversal, Exterior angles, Interior angles, Interior opposite angles, Alternate angles, Corresponding angles**

If a straight line cuts two other straight lines, the former straight line is called the transversal of those two straight lines.

In the image, the straight line EF cuts the two straight lines AB and CD.

Therefore, the straight line EF is the transversal of the straight lines AB and CD.

When a straight line cuts two other straight lines then eight angles are formed. Among them four angles are in the inside region of the two straight lines. These four angles are called interior angles and the other four angles are called exterior angles.

**Wbbse Class 7 Maths Solutions**

In the image, the interior angles are ∠AGH, ∠GHC, ∠GHD, and ∠HGB (or ∠3, ∠6, ∠5, and ∠4). The exterior angles are ∠EGA, ∠EGB, ∠CHF, and ∠FHD (or ∠2, ∠1, ∠7, and ∠8).

The further off interior angle, in respect of an exterior angle is called interior opposite angle.

For example, ∠GHD (or ∠5) is the interior opposite angle in respect of ∠EGB (or ∠1).

The interior angle adjacent to one exterior angle and the interior angle adjacent to further off interior angle in respect of the same exterior angle are called alternate angles to each other.

In the image, ∠GHD{or ∠5) is the alternate angle of ∠AGH (or∠3), ∠GHC (or∠6) is the alternate angle of ∠BGH (or ∠4).

An exterior angle and an interior opposite angle on the same side of the transversal are called corresponding angles.

In the images, the pair of angles [∠EGB (or ∠1), ∠GHD(or ∠5)], [∠EGA(or ∠2), ∠GHC(or ∠6), [∠AGH(or ∠3), ∠CHF(or ∠7)] and [∠BGH (or∠4), ∠DHF(or ∠8)] are corresponding angles.

**Interior angles on the same side of the transversal**

Obviously, ∠BGH and ∠GHD (i.e., ∠4 and ∠5) and also ∠AGH and ∠GHC (i.e., ∠3 and ∠6) are the interior angles on the same side of the transversal.

**Parallel Lines and Transversal**

Two lines in the same plane are said to be parallel lines if the perpendicular distance between them is the same at every intermediate pair of points when produced indefinitely in either direction.

The adjacent imges shows that AB and CD are two parallel lines.

The perpendicular distance between them at every intermediate pair of points (two perpendicular distances PQ and XY are shown in the image) is same.

If the two parallel lines are extended indefinitely in both directions, then also it would be found that the perpendicular distance between them remains same.

A pair of railway tracks is the most common example of parallel lines.

A pair of parallel straight lines (AB and CD) and their transversal EF are shown in the adjacent image.

Since the two straight lines AB and CD are parallel to each other, it can be proved that:

- Pair of corresponding angles are equal to each other. Therefore, in this case, ∠1 = ∠5, ∠2 =∠6, ∠3 = ∠7 and ∠4 = ∠8.
- Alternate angles are equal to each other. Therefore, in this case, ∠3 = ∠5 and ∠4 = ∠6.

Taking into account both (1) and (2) we can say that, ∠1 = ∠3 = ∠5 = ∠7 and ∠2 =∠4 = ∠6 = ∠8.

3. The sum of the interior angles on the same side of the transversal is 180° or two right angles.

∴ In this case, ∠3 +∠6 = 180° and ∠4 + ∠5 = 180°.

If a transversal crosses the parallel straight lines at right angles, it is called a perpendicular transversal.

In the adjacent image shown, XY is the perpendicular transversal to the pair of parallel lines AJB and CD.

It is clear from the imagethat, ∠3 + ∠6 = 180° (or 2 right angles) and ∠4 + ∠5 = 180° (or 2 right angles).

**Class Vii Math Solution Wbbse**

**Some examples **

**Example 1. What is the sum of the angles at a point? **

Solution:

**Sum of the angles at a point:**

An angle is measured with reference to a circle with its centre at the common endpoint of the rays. Hence, the sum of angles at a point is always 360°. It is called a full angle (see image).

**Example 2. **What are the values of smallest and largest geometric angles?

Solution: Smallest value = 0°, largest value = 360° (full angle or four right angles).

**Example 3.****Identify the acute, obtuse and reflex angles among the following: 45°, 72°, 187°, 210°, 175°, 300°, 15°, 120°, 140°**

Solution:

**Acute angles:** 15°, 45°, 72°

**Obtuse angles:** 120°, 140°, 175°.

**Reflex angles:** 187°, 210°, 300°.

**Example4. **

- What are the values of complementary and supplementary angles to an angle θ?
- What are the values of complementary and supplementary angles to 0°?
- Which angle is equal to its complementary angle?
- Which angle is equal to its supplementary angle?

**Class Vii Math Solution Wbbse**

Solution:

- The complementary and supplementary angles to θ are (90°-θ) and (180°-θ) respectively.
- Complementary angle to 0°= 90° – 0° = 90°. Supplementary angle to 0° = 180° – 0° = 180°
- If the required angle be x°,then according to the question the complementary angle shall be also x°.∴ x + x = 2x = 90° or, x = 90°/2 = 45
- If the required angle be x°,then according to the question the supplementary angle shall be also x°.∴ x + x = 2x = 180° or, x =180°/2 = 90°

**Example 5. **An angle is equal to twice its supplement. Determine its measure.

Solution: Let the required angle be x°.

According to the question, the supplementary angle = x°/2

∴ x + \(\frac{x}{2}\) = 180°

or, 2x + x = 360° or, 3x = 360° or x = 120°.

**Example 6. Look at the adjacent image. For what value of x, the points A, O, and B shall lie on the same straight line?**

Solution:

For A, O and B to lie on same straight line, the sum of ∠AOC and ∠BOC has to be 180°, i.e., ∠AOC + ∠BOC =180°.

∴ (6x + 5) + (4x-25) = 180

or, 10x – 20 – 180 or, 10x = 200 or, x = 20

**Example 7. In the image, ∠AOC = 50°. Find out the values of other angles.**

** **** Solution**:

Vertically opposite angles are equal to each other.

∴ ∠AOC = ∠BOD** **= 50°.

The sum of the angles at point 0 is 360° ∠COB** + ∠**AOD

**– 360° – (50° + 50°) = 260°**

Since these are vertically opposite angles, ∠COB = ∠AOD** **= 130°.

**Example 8. Indicate two pairs of alternate angles in the image shown.**

** Solution**: Considering PQ

**as transversal, ∠AEG**

**and ∠EGH**

**are alternate angles.**

Considering RS** **as transversal, ∠BFH

**and ∠FHG**

**are alternate angles.**

**Example 9. Parallel lines are shown in the images. Find the measure of angles indicated by x.**

Solution: See the image.

Therefore, x+ 60° = 180° or, x** **= 120°

The sum of the two interior angles on the same side of the transversal is 180°.

∴ x + 140° = 180° or, x** **= 40°.

**Class Vii Math Solution Wbbse**

** Example 10. **In the given image if AB//CD, find x, y

**and z.**

Solution:

∠x = 45° (Vertically opposite angle).

∠x =∠y** **= 45° (Alternate angle)

Since AB//CD,

∠z + ∠y** **= 180°.

or ∠z – 180° – ∠y – 180° – 45° = 135°.

**Classification of triangle**

**Triangle:** A plane figure bounded by three line segments is called a triangle. A triangle has three sides and three angles. Three angular points are called the vertices of the triangle.

ABC** **is a triangle. Its three sides are AB, BC,

**and AC.**

**Its three angles are ∠ABC, ∠BCA,**

**and ∠CAB.**

**Its three vertices are A, B,**

**and C.**

The sum of the three angles of a triangle is equal to 2 right angles or 180°.

In any triangle, the sum of any two sides is always greater than the third side. Again, the difference of any two sides is always less than the third side.

If any angular point of a triangle is taken as a vertex then its opposite side is called its base. The angle opposite to the base of a triangle is called its vertical angle.

If BC** **is taken as a base then ∠BAC

**will be its vertical angle.**

On the basis of sides there are three types of triangles: equilateral triangle, isosceles triangle, and scalene triangle.

**Class Vii Math Solution Wbbse**

**Equilateral triangle:** If the lengths of the three sides of a triangle are same then the triangle is called an equilateral triangle.

**Isosceles triangle:** If the lengths of the two sides of a triangle are same then the triangle is called an isosceles triangle.

**Scalene triangle:** If the lengths of the three sides of a triangle are unequal then the triangle is called a scalene triangle.

**On the basis of angles there are three types of triangles:** acute-angled triangle, obtuse-angled triangle, and right-angled triangle.

**Acute-angled triangle:** If each of the three angles of a triangle is acute then the triangle is called an acute angled triangle.

**Obtuse-angled triangle:** If any one angle of a triangle is obtuse then the triangle is called an obtuse-angled triangle.

**Right-angled triangle:** If any one angle of a triangle is a right angle then the triangle is called a right-angled triangle.

**Median of a triangle**

The line segment obtained by joining the middle point of any side of a triangle to the A opposite vertex is called the median of the triangle.

In the image, the line segment AD has been obtained by joining the mid point D of the side BC of the triangle ABC to the opposite vertex A.

Hence, AD is a median of the triangle ABC. There are always three medians of a triangle.

The three medians of a triangle always intersect at the same point, i.e., they are concurrent.

Therefore, in each case of a triangle like scalene or equilateral or isosceles, or acutely angled or obtuse-angled or right-angled triangle, the three medians are concurrent.

**Class Vii Math Solution Wbbse**

The point of concurrence of the medians is known as the centroid of the triangle.

In the image, AD, BE and CF are the three medians of ΔABC.

They intersect at point G. G is the centroid of ΔABC.

**Height of a triangle**

The line segment obtained by drawing the perpendicular from any vertex of a triangle to the opposite side is called the height of the triangle.

In the image, AD has been drawn perpendicular from the vertex A to the opposite side BC of the triangle ABC.

Hence, AD is the height of the triangle ABC. In this case BC is the base of the triangle.

If AC is taken as the base of the triangle then the perpendicular from B on AC will be the height of the triangle.

If AB is taken as the base of the triangle then the perpendicular from C on AB will be the height of the triangle.

**Some examples **

**Example 1. Find the measures of the angles of an equilateral triangle.**

Solution:

Each angle is equal to 60° (see the adjacent image)

**Example 2. What are the measures of the three angles of a right-angled isosceles triangle?**

Solution:

The measures of the three angles of a right-angled isosceles triangle are 45°, 45°, and 90°. See the adjacent image.

In this case, AB = AC and ∠BAC = 90°.

**Example 3. Find the maximum and maximum number of acute in a triangle.**

Solution: The minimum number of acute angles is two and the maximum number is three.

**Example ****4. **What is the nature of three points through which it is not possible to construct a triangle?

Solution: It is not possible to construct a triangle through three points when they are collinear.

**Example ****5. **How many parts of a triangle should be known to construct it?

Solution: Three.

[Of course, it is not possible to construct a definite triangle when three angles of it are known. Because when the three angles are given then it should be assumed that actually two angles are given. If two angles of a triangle are known then automatically the third one is known. In other cases, e.g., when three sides or two sides and one angle are known, it is possible to construct a definite triangle.]

**Example ****6. **How many straight lines may be drawn through a point?

Solution: An infinite number of straight lines may be drawn through a point.

**Example ****7. **How many straight lines may be drawn through two points?

Solution: One straight line.

**Example 8. **What is the minimum number of curved lines required to enclose a region?

Solution: One.

**Example ****9. **What is the minimum number of straight lines required to enclose a region?

Solution: Three.

**Example ****10. **If the sum of the two angles of a triangle be 90° then what is its name?

Solution: As the sum of the two angles is 90°, the third angle must be equal to 90°. Hence the triangle is a right-angled triangle.

**Example 11.** **A triangle is equilateral on the basis of sides. What is its name on the basis of angles?**

Solution: Acute angled.

**Example 12.****Is it possible to construct a triangle having sides of lengths 3 cm, 4 cm, and 8 cm? **

Solution: We know that the sum of the two sides of a triangle is always greater than the third.

But in this case, 3 cm + 4 cm = 7cm. It is less than 8 cm. So, it is not possible to construct a triangle in this case.

**Example ****13. **Is it possible to construct a triangle having angles of measures 55°, 64°, and 60°?

Solution: We know that the sum of the three angles of a triangle is 180°.

But in this case the sum of the three angles is (55^{o}+64^{o}+60°) = 179°. So it is not possible to construct a triangle in this case.

**Example ****14. **If the sum of the two acute angles of an obtuse-angled triangle be 50° then find the measure of the other angle.

Solution: The measure of the other angle of the obtuse-angled triangle = 180° – 50° = 130°.

**Example **** 15. How many right angles may be there in a triangle**?

Solution: One

**Example ****16. **If one acute angle of a right-angled triangle be 40°, then find its other acute angle.

Solution: 50°.

**Class Vii Math Solution Wbbse**

**Example ****17. **When do the two straight lines lying on a plane intersect at a point?

Solution: When they are non-parallel.

**Example ****18. **What is the number of acute angles of a right-angled triangle?

Solution: Two.

**Example **** 19. **What is the number of medians of a triangle?

Solution: Three.

**Example 20. **If the angles of a triangle bear a ratio of 3:4:5, then determine the measure of the angles.

Solution:

**Given:**

The angles of a triangle bear a ratio of 3:4:5

Let us assume that the angles be 3x, 4y,** **and 5z

**.**∴ 3x + 4x + 5x = 180°

[ ∴ Sum of three angles of a triangle is 180°]

or 12x = 180° or, x = \(\frac{180}{2}\) = 15.

∴ The angles of the triangle will be (3x 15)°, (4×15)° and (5×15)° or 45°, 60° and 75°.

**Example 21.****Prove that if the measure of one angle of a triangle is equal to the sum of the rest two angles then the triangle is a right-angled one.**

Solution:

Let us assume that the three angles of the triangle be ∠A, ∠B, and ∠C.

According to the question, let us assume that ∠A = ∠B + ∠C.

Now ∠A + ∠B + ∠C = 180°

or ∠A + ∠A = 180° (Since ∠A = ∠B + ∠C assumed) or, 2 ∠A = 180°

or ∠A = 180°/2 = 90° .

Therefore the triangle is a right triangle. (Proved)

**Example 22. **In the given image, DEIIBC. If ∠A = 65° and ∠B = 50°, then determine the values of ∠ADE, ∠AED, and∠C.

Solution:

**Given:**

In the given image, DEIIBC. If ∠A = 65° and ∠B = 50°

DE//BC and ADB is the transversal

∴ ∠ADE = ∠DBC = ∠B = ∠50° (corresponding angles)

The sum of three angles in ΔADE is 180°.

**Classification of quadrilateral**

**Quadrilateral:**

A plane figure enclosed by four line segments is called a quadrilateral. A quadrilateral has four sides and four angles.

Four angular points are called the four vertices of a quadrilateral.

The line segment joining any two opposite vertices of a quadrilateral is called the diagonal of the quadrilateral.

There are two diagonals of a quadrilateral. ABCD is a quadrilateral. Its four sides are

AB, BC, CD, and DA. Its four angles are ∠ABC, ∠BCD, ∠CDA, and ∠DAB.

Its four angular points are A, B, C, and D. Its two diagonals are AC and BD.

**Parallelogram:** If the opposite sides of a quadrilateral are parallel then it is called a parallelogram.

In the image shown is a parallelogram ABCD since the opposite sides of the quadrilateral are parallel to each other, i.e., ADIIBC and ABIIDC

∴ AB = DC and AD = BC.

The diagonals of a parallelogram bisect each other.

**Class Vii Math Solution Wbbse**

∴ AO = OC and BO = OD. The opposite angles of a parallelogram are equal to each other.

∴ ∠BAD = ∠BCD and ∠ADC = ∠ABC.

Since the opposite sides of a parallelogram are parallel, hence the two angles adjacent to any side are supplementary to each other.

** Rectangle:** If one angle of a parallelogram is a right angle then it is called a rectangle. The diagonals of the rectangle are equal in length and they bisect each other.

** Square:** If the lengths of the two adjacent sides of a rectangle are equal then it is called a square.

Therefore all the sides of a square are of equal length and each angle is a right angle.

The diagonals of a square are equal in length and they bisect each other perpendicularly.

**Trapezium:** If only one pair of opposite sides of a quadrilateral are parallel then it is called a trapezium.

In trapezium ABCD, ABIIDC. The nonparallel sides of the trapezium are known as the oblique sides.

**Isosceles trapezium:** If the lengths of the non-parallel sides (i.e., oblique sides) of a trapezium are equal then it is called an isosceles trapezium. In the adjacent image, ABCD is an isosceles trapezium since AD = BC.

**Rhombus:** If the lengths of the four sides of a quadrilateral are equal but none of the angles is a right angle then it is called arhombus. The image ABCD is a rhombus in which AB = BC – CD – DA; but none of its angles is a right angle.

The opposite sides of a rhombus are parallel to each other. The diagonals of a rhombus are unequal in length, but they bisect each other perpendicularly. Therefore, AC ≠ BD, but AO = OC and BO = OD.

**Kite:** A quadrilateral in which two pairs of adjacent sides are equal is called a kite.

In the image, AB = AD and CD = CB. Thus adjacent sides of each pair are equal. The diagonals of a kite are perpendicular to each other.

**Some examples ****Example **

**Example 1. How many parts of a quadrilateral should be known to construct it?**

Solution: Five.

**Example 2. ****In this quadrilateral, there is only one pair of parallel sides.**

Solution: Trapezium.

**Class Vii Math Solution Wbbse**

** ****Example 3. **Which quadrilaterals have sides of equal lengths?

Solution: Square and Rhombus.

**Example 4. **By what name do we call a parallelogram whose one angle is a right angle?

Solution: Rectangle.

**Example 5. **How many diagonals and how many vertices are there in a quadrilateral?

Solution: Two diagonals and four vertices.

**Example ****6. **What is the sum of the angles of a quadrilateral?

Solution: 360°.

**Example 7. **What is the difference between the two diagonals of a rectangle and the two diagonals of a rhombus?

Solution: The two diagonals of a rectangle are equal and the two diagonals of a rhombus are unequal.

**Example ****8.****What kind of quadrilateral is it whose all the angles are equal but adjacent sides are unequal?**

Solution: Rectangle.

**Example 9. What kind of quadrilateral is it whose diagonals are unequal in length but are perpendicular to each other?**

Solution: Rhombus.

**Example ****10. **Name the quadrilaterals whose diagonals are equal to each other in length.

Solution: Rectangle, square and isosceles trapezium.

**Example **11. What are the measures of ∠B and ∠C in the parallelogram ABCD in which ∠A = 75°?

Solution:

**Given:**

∠A = 75°

The opposite angles of a parallelogram are equal to each other.

∴ ∠A = ∠C = 75°.

In a parallelogram, the two angles adjacent to each side are supplementary to each other.

∴ ∠B+ ∠C= 180° (Both are adjacent to BC)

or, ∠B = 180° – ∠C = 180° – 75° = 105°

The measures of ∠B and ∠C in the parallelogram ABCD ∠B = 180° – ∠C = 180° – 75° = 105°

**Example 12. **If a pair of opposite angles of a parallelogram be 2x – 50° and x + 20°, then what type of parallelogram is it?

Solution:

**Given:**

If a pair of opposite angles of a parallelogram be 2x – 50° and x + 20°

The opposite angles are equal to each other in a parallelogram.

∴ 2x – 50 = x + 20 or, x = 70

Putting the value of x** **we get, (2 x 70) – 50 = 90° and 70 + 20 = 90°, i.e., each of the angles of this parallelogram is a right angle.

∴ The parallelogram in question is a rectangle.