# ISOSCELES EQUILATERAL AND SCALENE TRIANGLE WORKSHEET

Problem 1 :

Use the diagram of ΔABC shown below to prove the Base Angles Theorem.

Problem 2 :

In the diagram shown below,

(i) find the value of x

(ii) find the value of y

Problem 3 :

In the diagram shown below,

(i) find the value of x

(ii) find the value of y

Problem 1 :

Use the diagram of ΔABC shown below to prove the Base Angles Theorem.

Solution :

Given : In ΔABC, AB  ≅  AC

To prove : ∠B  ≅  ∠C

Proof :

(i)  Draw the bisector of ∠CAB.

(iii)  We are given that AB  ≅  AC. Also DA  ≅  DA, by the Reflexive Property of Congruence.

(iii) Use the SAS Congruence Postulate to conclude that ΔADB ≅ ΔADC.

(iv)  Because corresponding parts of congruent triangles are congruent, it follows that ∠B  ≅  ∠C.

Problem 2 :

In the diagram shown below,

(i) find the value of x

(ii) find the value of y

Solution (i) :

In the diagram shown above, 'x' represents the measure of an angle of an equilateral triangle.

From the corollary given above, if a triangle is equilateral, then it is equiangular.

So, the measure of each angle in the equilateral triangle is x.

By the Triangle Sum Theorem, we have

x° + x° + x°  =  180°

Simplify.

3x°  =  180°

3x  =  180

Divide both sides by 3 to solve for x.

x  =  60

Solution (ii) :

In the diagram shown above, the vertex angle forms a linear pair with a 60° angle, so its measure is 120°.

It has been illustrated in the diagram given below.

By the Triangle Sum Theorem, we have

120° + 35° + y°  =  180°

Simplify.

155° + 2y°  =  180°

155 + 2y  =  180

Subtract 155 from both sides.

y  =  25

Problem 3 :

In the diagram shown below,

(i) find the value of x

(ii) find the value of y

Solution (i) :

In the diagram shown above, 'x' represents the measure of an angle of an equilateral triangle.

From the corollary given above, if a triangle is equilateral, then it is equiangular.

So, the measure of each angle in the equilateral triangle is x.

By the Triangle Sum Theorem, we have

x° + x° + x°  =  180°

Simplify.

3x°  =  180°

3x  =  180

Divide both sides by 3 to solve for x.

x  =  60

Solution (ii) :

In the diagram shown above, 'y' represents the measure of a base angle of an isosceles triangle.

From the Base Angles Theorem, the other base angle has the same measure. The vertex angle forms a linear pair with a 60° angle, so its measure is 120°.

It has been illustrated in the diagram given below.

By the Triangle Sum Theorem, we have

120° + y° + y°  =  180°

Simplify.

120° + 2y°  =  180°

120 + 2y  =  180

Subtract 120 from both sides.

2y  =  60

Divide both sides by 2.

y  =  30

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