TRIANGLE SUM THEOREM WORKSHEET

1.  Prove that the sum of the measures of the interior angles of a triangle is 180°. 

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

3.  Find the missing angles in the triangle shown below. 

4.  The measure of one acute angle of a right triangle is two times the measure of the other acute angle. Find the measure of each acute angle. 

5.  Find the missing angles in the triangle shown below. 

6.  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. 

7)  

8)  In a triangle ABC, ∠ A = 2X+7, ∠ B = 5X-9, and ∠ C = 6X. What is the value of x and what are the measures of angles A, B, and C?

9)  Two angles of a triangle have equal measures, but the third angle's measure is 36° less than the sum of the other two. Find the measure of each angle of the triangle.

10)  Find the measures of the angles of an isosceles triangle if the measure of the vertex angle is 40 degrees less than the sum of the measures of the base angles

1. Answer :

Given :

Triangle ABC. 

To Prove :

 m∠1 + m∠2 + m∠3  =  180°

Plan for Proof : 

By the Parallel Postulate, we can draw an auxiliary line through point B and parallel to AC. Because ∠4, ∠2 and ∠5 form a straight angle, the sum of their measures is 180°.

We also know that ∠1 ≅ ∠4 and ∠3 ≅ ∠5 by the Alternate Interior Angles Theorem.   

2. Answer :

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

 30° +  60° + 90°  =  180°

The sum of the three angles is equal 180°. By Triangle Sum Theorem, the given three angles can be the angles of a triangle. 

3. Answer : 

In the triangle shown above, two sides are congruent. Angles opposite to congruent sides are always congruent. 

So, if one missing angle is assumed to be x°, then the other missing angle also must be x°. Because the two angles are congruent. 

The diagram shown below illustrates this. 

By Triangle Sum Theorem, the sum of the measures of the interior angles of a triangle is 180°. 

So, we have 

x° + x° + 40°  =  180°

Simplify. 

2x + 40  =  180

Subtract 40 from both sides. 

2x  =  140

Divide both sides by 2. 

x  =  70 

So, the measure of each missing angle is 70°.

4. Answer : 

Let A, B and C be the vertices of the triangle and right angle is at C. 

Let ∠A = x°, then ∠B = 2x°. The diagram shown below illustrates this. 

By Corollary to the Triangle Sum Theorem, the acute angles of a right triangle are complementary.  

So, we have 

x° + 2x°  =  90°

Simplify.

3x°  =  90°

Divide both sides by 3.

x  =  30

So, m∠A = 30°  and m∠B = 2(30°)  =  60°

So, the two acute angles are 30° and 60°. 

5. Answer :

In the triangle shown above, two sides are congruent. Angles opposite to congruent sides are always congruent. 

So, if one missing angle is assumed to be x°, then the other missing angle also must be x°. Because the two angles are congruent. 

The diagram shown below illustrates this. 

In the triangle shown above, one of the angles is right angle. So, it is right triangle.

By Corollary to the Triangle Sum Theorem, the acute angles of a right triangle are complementary.  

So, we have 

x° + x°  =  90°

Simplify. 

2x  =  90

Divide both sides by 2. 

x  =  45

So, the measure of each missing angle is 45°.

6. Answer :

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°

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

7. Answer :

<ADE = 6x - 10(Vertical angles)

<DEC + <DEA = 180

95 + <DEA = 180

<DEA = 180 - 95

<DEA = 85

<ADE + <DEA + <DAE = 180

6x - 10 + 85 + 4x + 5 = 180

10x + 90 - 10 = 180

10x + 80 = 180

10x = 180 - 80

10x = 100

x = 100/10

x = 10

8. Answer :

∠A = 2X + 7, ∠B = 5X - 9, and ∠C = 6X

Sum of interior angles = 180

2x + 7 + 5x - 9 + 6x = 180

13x - 2 = 180

13x = 180 + 2

13x = 182

x = 14

9. Answer :

Let x be the equal angles.

Third angle = 2x - 36

x + x + 2x - 36 = 180

4x - 36 = 180

4x = 180 + 36

4x = 216

x = 216/4

x = 54

2x - 36 = 2(54) - 36 ==> 72

So, the three angle measures are 54, 54 and 72.

10. Answer :

Let x be the measures of isosceles triangle.

vertex angle = 2x - 40

x + x + 2x - 40 = 180

4x - 40 = 180

4x = 180 + 40

4x = 220

x = 220/4

x = 55

2x - 40 = 2(55) - 40

= 110 - 40

= 70

So, the three angle measures are 55, 55 and 70

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.