Finding Angles

Intersecting Lines

1.  Consecutive angles are supplementary: Angles 1 and 2 add to 180o.

2.  Vertical angles are congruent:  Angles 1 and 3 are congruent.

EXAMPLE 1:  If angle BPC is four times as large as angle CPD, then what is the measure of angle APB?

SOLUTION:  If x is the measure of angle CPD, then 4x is the measure of angle BPC:

Angles BPC and CPD are supplementary, so 4x + x = 180.  Therefore 5x = 180, so x = 36.   That is, angle CPD measures 36o. Angles APB and CPD are vertical angles and therefore congruent.  So angle APB measures 36o.

EXAMPLE 2:  Find the values of x and y:

SOLUTION:

Using vertical angles, we know that 2x – 8 = –2(4 – x)

The right side simplifies to –8 + 2x or 2x – 8 so this equation is an identity:

2x – 8 = 2x – 8

Any number makes this true, so we cannot find x directly.

Again by vertical angles, 2y + 4 = 3y – 64.  This equation is not an identity and we can solve for y

4 = y – 64

68 = y

So we know y = 68, and therefore the two angles containing ymeasure 2(68) + 4 = 140o.

Since the angle marked 2x – 8 is supplementary to this angle, we now know

2x – 8 = 140, so 2x = 148, and therefore x = 74.

Parallel Lines

When two parallel lines are cut by a transversal, there are 8 pairs of congruent angles and 8 pairs of supplementary angles.  This can be understood by identifying a pair of corresponding angles, and using vertical angles and linear pairs.  Corresponding angles can be thought of as angles that match when a picture of one angle at the intersection of one of the parallel lines with the transversal is moved to the other parallel line:

The following summarizes angle congruences for two parallel lines cut by a transversal:

All other pairs of angles (that is, pairs marked with an even and an odd number) are supplementary.

EXAMPLE 3:  Find x and y if lines a and b are parallel, and determine if lines c and d are parallel.

SOLUTION:  Since a and b are parallel, 2x + 13 = 3x – 24 (they are alternate interior angles).  Solving gives x = 37.

These angles measure 2 . 37 + 13 = 87 degrees, and the angle marked 3y + 24 is supplementary to this, so 3y + 24 = 93.

Solving this equation gives y = 23.

Then 4y – 5 = 92 – 5 = 87, which again is supplementary to 3y + 24.

Since these are same-side interior angles for lines c and d cut by transversal a, lines c and d are parallel.

Parallelograms

A parallelogram is a quadrilateral in which opposite sides are parallel.  Pairs of interior angles are either congruent or supplementary:

Opposite angles are congruent (angles 1 and 3, and angles 2 and 4).

Consecutive angles are supplementary (angles 1 and 2, angles 2 and 3, angles 3 and 4, and angles 1 and 4).

EXAMPLE  4:  Could the following figure be a parallelogram?

SOLUTION:  If it is a parallelogram, then 5x = 3x + 46 since opposite angles are congruent.

Solving gives x = 24, so 5x = 120.

If it is a parallelogram, then 5y = 180 – 120 = 60 since consecutive angles are supplementary, so y = 12.

Therefore 3y + 28 = 36 + 28 = 64.  But the angles marked 5y and 3y + 28 should be congruent, which is not the case.

Therefore this cannot be a parallelogram.

Triangles

The main results about triangles are these:

1.  The angles of a triangle add to 180o.

2.  An exterior angle of a triangle equals the sum of the two remote interior angles:

3.  The base angles of an isosceles triangle are congruent, and conversely:

4.  The side opposite the smallest angle in a triangle is the shortest side and the side opposite the largest angle is the longest side, and conversely.

EXAMPLE 5:

One angle of a triangle is twice as large as another angle, and the third angle is 20o less than the largest angle.  What are the measures of the angles in this triangle?

SOLUTION:  We can let x and 2x denote the measures of two angles.

Then 2x – 20 is the measure of the third angle.

Since the angles add to 180o, we can write the following equation:

(x) + (2x) + (2x – 20) = 180

Solving:  5x – 20 = 180,    5x = 200,  x = 40

The angles are:

x = 40o

2x = 80o

2x – 20 = 60o

EXAMPLE 6:  Find the measure of angle ABC:

SOLUTION:

The external angle is the sum of its two remote interior angles, so we can write the following equation:

4x + 24 = 2x + 3x

Solving, we find that x= 24.

Therefore the exterior angle measures 4 . 24 + 24 = 120o

Angle ABC is supplementary to this, so it measures 180 – 120 = 60o.

EXAMPLE 7:  Which side is longest in the following triangle?

SOLUTION:  First we solve for x:

4x + 14 + 5x + 4 + 6x – 3 = 180

15x + 15 = 180

15x = 165

x = 11

Now we find the angles:

angle A:  4x + 14 = 58o

angle B:  5x + 4 = 59o

angle C:  6x – 3 = 63o

Since angle C is the largest angle, side AB is the longest side.

Polygons

A polygon of n sides can be partitioned into n – 2 non-overlapping triangles with vertices in common with the polygon.  For example, a quadrilateral has 4 sides and can be split into 2 triangles, and a triangle has 3 sides and consists of 1 triangle.

Since the angles of a triangle add to 180 degrees, the angles of an n-sided polygon must add to 180(n – 2) degrees.

EXAMPLE 8:  Find the measure of angle A in the following pentagon:

SOLUTION:

A pentagon has 5 sides, so can be partitioned into 3 triangles:

Therefore its angles add to 3 times 180o, so

(2x + 3) + (3x) + (x + 3) + (3x – 5) + 2x = 3 . 180

11x + 1 = 540

11x = 539

x = 49

So angle A measures 2 . 49 + 3 = 101o