TRIANGLES AND ANGLES

A triangle is a figure formed by three noncollinear points. 

When the dies of a triangle are extended, other angles are formed. The three original angles are the interior angles. The angles that are adjacent to the interior angles are the exterior angles.

Each vertex has a pair of congruent exterior angles. It is common to show only one exterior angle at each vertex.   

Triangle Sum Theorem

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

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. 

Corollary to the Triangle Sum Theorem

The acute angles of a right triangle are complementary.  

Solving Problems on Triangles and Angles

Problem 1 :

Can the following angle measures be the angles of a triangle?

39°, 41° and 100° 

Solution :

In a triangle, the three angles always add up to 180°.

Add the given angle measures :

30° + 60° + 90° = 180°

Since the given three angle measures add up to 180°, they can be the angles of a triangle.

Problem 2 :

In a right triangle, if the measure of one acute angle is equal to two times the measure of the other acute angle, find the measure of each acute angle.

Solution :

Consider A, B and C as the vertices of the triangle and assume right angle is at C.

If ∠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.

x° + 2x° = 90°

Simplify.

3x° = 90°

Divide both sides by 3.

x = 30

m∠A = 30°

m∠B = 2(30°) = 60°

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

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 :

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)°

x + 65 = 2x + 10

Subtract x from both sides.

65 = x + 10

Subtract 10 from both sides.

55 = x

So, the value of x is 55.

Problem 5 :

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°

2x  =  90

Divide both sides by 2.

x = 45

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

Problem 6 :

The measure of first angle of a triangle is 15° more than than the measure of second angle and the measure of third angle is 30° less than the measure of second angle. Find the measures of three angles of the triangle.

Solution :

Let x° be the measure of first angle.

The measure of second angle :

= (x + 15)° ----(1)

The measure of third angle :

= (x + 15)° - 30°

= x° + 15° - 30°

= x° - 15°

= (x - 15)° ----(2)

In a triangle, the three angles always add up to 180°.

x° + (x + 15)° + (x - 15)° = 180°

x + x + 15 + x - 15 = 180

3x = 180

Divide both sides by 3.

x = 60

first angle = 60°

Substitute x = 60 in (1) and (2) to get the measures of second and third angles.

second angle = (60 + 15)° = 75°

third angle = (60° - 15)° = 45°

So, the measures of three angles of the triangle are

60°, 75° and 45°

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