**Problem 1 :**

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

**Problem 2 :**

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

**Problem 3 :**

Find the missing angles in the triangle shown below.

**Problem 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.

**Problem 5 :**

Find the value of x in the diagram shown below.

**Problem 6 : **

Find the missing angles in the triangle shown below.

**Problem 1 : **

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

**Solution : **

**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.

Draw BD parallel to AC m∠4 + m∠2 + m∠5 = 180° aaaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaa ∠1 ≅ ∠4 and ∠3 ≅ ∠5 aaaaaaaaaaaaaaaaaa m∠1 = m∠4 and m∠3 = m∠5 aaaaaaaaaaaaaaaaa m∠1 + m∠2 + m∠3 = 180° aaaaaaaaaaaaaaaa |
Parallel Postulate Angle addition postulate and definition of straight angle. Alternate Interior Angles Theorem Definition of congruent angles. Substitution property of equality. |

**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°

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

**Problem 3 :**

Find the missing angles in the triangle shown below.

**Solution : **

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°.

**Problem 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.

**Solution : **

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°.

**Problem 5 :**

Find the value of x in the diagram shown below.

**Solution :**

By Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

So, we have

x° + 65° = (2x + 10)°

or

x + 65 = 2x + 10

Subtract x from both sides.

65 = x + 10

Subtract 10 from both sides.

55 = x

**Problem 6 : **

Find the missing angles in the triangle shown below.

**Solution : **

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°.

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