# ANGLE RELATIONSHIPS IN PARALLEL LINES AND TRIANGLES

## About "Angle relationships in parallel lines and triangles"

Angle relationships in parallel lines and triangles :

In this section, we are going to see the angle relationships, when two parallel lines are cut by a transversal and in triangles.

## Angle Pairs Formed by a Transversal Corresponding angles :

Angles lie on the same side of the transversal t, on the same side of lines a and b.

Example : ∠ 1 and ∠ 5

Alternate interior angles :

Angles are nonadjacent angles that lie on opposite sides of the transversal t, between lines a and b.

Example : ∠ 3 and ∠ 6

Alternate exterior angles :

Angles lie on opposite sides of the transversal t, outside lines a and b.

Example : ∠ 1 and ∠ 8

Same-side interior angles :

Angles lie on the same side of the transversal t, between lines a and b.

Example : ∠ 3 and ∠ 5

## Angle relationships in triangles

In this section, we are going to see the angle relationships in triangles through the following steps.

Step 1 :

Draw a triangle and cut it out. Label the angles A, B, and C. Step 2 :

Tear off each “corner” of the triangle. Each corner includes the vertex of one angle of the triangle.

Step 3 :

Arrange the vertices of the triangle around a point so that none of your corners overlap and there are no gaps between them. Step 4 :

What do you notice about how the angles fit together around a point ?

The angles form a straight angle.

Step 5 :

What do you notice about how the angles fit together around a point ?

180°

Step 6 :

Describe the relationship among the measures of the angles of triangle ABC ?

The sum of the angle measures is 180°.

Step 7 :

What does the triangle sum theorem state ?

The triangle sum theorem states that for triangle ABC,

mA + mB + mC =  180°

## Exterior angle theorem in triangles The theorem states that the measure of an exterior  angle is equal to the sum of its  remote interior angles.

That is,

m∠1 + m∠2  =  m∠4

## Angle relationships in parallel lines and triangles - Problems

Problem 1 :

In the figure given below,  let the lines l₁ and l₂ be parallel and m is transversal. If F  =  65°, find the measure of each of the remaining angles. Solution :

From the given figure,

F and H are vertically opposite angles and they are equal.

Then, H  =  F -------> H  =  65°

H and D are corresponding angles and they are equal.

Then, D  =  H -------> D  =  65°

D and B are vertically opposite angles and they are equal.

Then, B  =  D -------> B  =  65°

F and E are together form a straight angle.

Then, we have

F + E  =  180°

Plug F  =  65°

F + E  =  180°

65° + E  =  180°

E  =  115°

E and G are vertically opposite angles and they are equal.

Then, G  =  E -------> G  =  115°

G and C are corresponding angles and they are equal.

Then, C  =  G -------> C  =  115°

C and A are vertically opposite angles and they are equal.

Then, A  =  C -------> A  =  115°

Therefore,

A  =  C  =  E  =  G  =  115°

B  =  D  =  F  =  H  =  65°

Let us look at the next problem on "Angle relationships in parallel lines and triangles".

Problem 2 :

Can 30°, 60° and 90° be the angles of a triangle ?

Solution :

Let us add all the three given angles and check whether the sum is equal to 180°.

30° +  60° + 90°  =  180°

Since the sum of the angles is equal 180°, the given three angles can be the angles of a triangle.

Let us look at the next problem on "Angle relationships in parallel lines and triangles".

Problem 3 :

In a triangle, if the second angle is 5° greater than the first angle and the third angle is 5° greater than second angle, find the three angles of the triangle.

Solution :

Let "x" be the first angle.

The second angle  =  x + 5

The third angle  =  x + 5 + 5  =  x + 10

We know that,

the sum of the three angles of a triangle  =  180°

x + (x+5) + (x+10)  =  180°

3x + 15  =  180

3x  =  165

x  =  55

The first angle  =  55°

The second angle  =  55 + 5  =  60°

The third angle  =  60 + 5  =  65°

Hence, the three angles of a triangle are 55°, 60° and 65°.

Let us look at the next problem on "Angle relationships in parallel lines and triangles".

Problem 4 :

Find m∠W and m∠X in the triangle given below. Solution :

Step 1 :

Write the Exterior Angle Theorem as it applies to this triangle.

m∠W + m∠X  =  m∠WYZ

Step 2 :

Substitute the given angle measures.

(4y - 4)° + 3y°  =  52°

Step 3 :

Solve the equation for y.

(4y - 4)° + 3y°  =  52°

4y - 4 + 3y  =  52

Combine the like terms.

7y - 4  =  52

7y - 4 + 4  =  52 + 4

Simplify.

7y  =  56

Divide both sides by 7.

7y / 7  =  56 / 7

y  =  8

Step 4 :

Use the value of y to find m∠W and m∠X.

m∠W  =  4y - 4

m∠W  =  4(8) - 4

m∠W  =  28

m∠X  =  3y

m∠X  =  3(8)

m∠X  =  24

So, m∠W  =  28° and m∠X  =  24°. After having gone through the stuff given above, we hope that the students would have understood "Angle relationships in parallel lines and triangles".

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