Regents Examination in Geometry: Angle and Segment Relationships

Angle and Segment Relationships

In this section, we explore some of the basic relationships involving the measures of angles and segments. Algebraic modeling and equation solving will be applied to find the measure of a missing angle or the measure of a segment. These relationships represent the building blocks that will be used to explore more complex problems, theorems, and proofs.

BASIC ANGLE RELATIONSHIPS
Key Ideas
Basic theorems and definitions used to solve problems involving angles include:
The sum of the measures of adjacent angles around a point equals 360°.
Supplementary angles have measures that sum to 180°.
Complementary angles have measures that sum to 90°.
Vertical angles are the congruent opposite formed by intersecting lines and are congruent.
Angle bisectors divide angles into two congruent angles.
The measure of a whole equals the sum of the measures of its parts.

Sum of the Angles About a Point
The measures of the adjacent angles about a point sum to 360°. In Figure 2.1, m∠1 + m∠2 + m∠3 + m∠4 = 360°.

Sum of the angles about a point = 360°

Example
Find the measure of each angle in the accompanying figure.

Solution:


 

Supplementary Angles, Complementary Angles, and Linear Pairs
Supplementary angles are angles whose measures sum to 180°. The angles may or may not be adjacent. Two adjacent angles that form a straight line are called a linear pair. They are supplementary. Figure below shows a linear pair. Multiple adjacent angles around a line also sum to 180°, as shown in Figure 2.3.

Linear pair

Adjacent angles around a line
 

Complementary angles are angles whose measures sum to 90°. As with supplementary angles, complementary angles may or may not be adjacent.

Complementary angles are illustrated below.

Complementary angles

Example 1

intersects

at B. If m∠ABD = (4x + 8)° and m∠CBD = (2x + 4)°, find the measure of each angle.

Solution: The angles form a linear pair and are supplementary.

Example 2
In the accompanying figure, m∠1 = (x + 16)° and m∠2 = (3x + 6)°. Find the measure of each angle.

Solution: The angles are complementary.


Vertical Angles
When two lines intersect, four angles are formed. Each pair of opposite angles are congruent and are called vertical angles.

In this figure, ∠1 and ∠3 are vertical angles and are therefore congruent. Also, ∠2 and ∠4 are vertical angles and are therefore congruent. Vertical angles show up frequently in geometry. So always be on the lookout for them. Once you know the measure of any one of the four vertical angles formed by two intersecting lines, you can easily calculate the measures of the other three using the supplementary angle relationship for a linear pair.

Vertical angles

Example
In Figure above, m∠1 = 150°. Find the measure of each of the other angles.

Solution: Vertical Angles
m∠3 = m∠1
m∠3 = 150°

Vertical Angles
m∠2 = m∠4
m∠2 = 30°Linear Pair
m∠4 + m∠1 = 180°
m∠4 + 150° = 180°
m∠4 = 30°

BISECTORS, MIDPOINT, AND THE ADDITION POSTULATE


Key Ideas
Whenever a segment or an angle is divided into two or more smaller parts, the sum of the smaller parts equals the original part. This is called the segment or angle addition postulate. When the part is divided exactly in half, we say it is bisected and the two parts are congruent to each other. An angle bisector is a line, segment, or ray that divides an angle into two congruent parts. A segment bisector intersects a segment at its midpoint, which divides it into two congruent parts.


Segment and Angle Addition
Any segment can be divided by locating a point between the two endpoints, creating two new segments. The segment addition postulate states a segment’s length equals the sum of the two new segments formed by any point between the endpoints of the original segment. As shown in Figure below, point C is located on  between A and B. Therefore, AC + CB = AB.

Segment addition

Example 1
Point S is located between points R and T on
. If RS = (2x + 1), ST = (3x + 4), and RT = 35, find the lengths of RS and ST.

Solution:

You should always sketch the figure and write the segment addition postulate in terms of segment names. This will give you the opportunity to check the postulate against the figure. If something does not make sense, you can make a correction before going further. Once you have substituted in algebraic expressions, it is much more difficult to check whether the equation makes sense from a geometry point of view.
The angle addition postulate is similar to the segment addition postulate. The measure of an angle is equal to the sum of the two adjacent angles formed by an interior ray with its endpoint at the vertex of the original angle.

Example 2
m∠ABC = (6x − 7)°, m∠ABD = (3x + 3), and m∠CBD = (2x + 12). Find m∠ABC.

Solution:

Besides addition, we also have postulates for segment subtraction and angle subtraction. These are used in a manner similar to segment and angle addition.

Example 3
Write an expression for the length of EB in terms of lengths AB and AE.

Solution:



Angle Bisectors
A line, segment, or ray that divides an angle into two congruent angles is called an angle bisector. Figure 2.7 illustrates an angle bisector. Ray

bisects ∠ACB. The two congruent angles formed are ∠ACD and ∠BCD.

Angle bisector 

Example
 bisects ∠ABD. If m∠ABD = (8x − 12)° and m∠ABC = (3x + 4)°, find the value of x.

Solution:


Midpoints and Segment Bisectors
In segments, the point that divides a segment into two congruent segments is called a midpoint. Any line, segment, or ray that intersects a segment at its midpoint is called a segment bisector, or simply bisector. Figure 2.8 illustrates a segment bisector. Line m bisects
 at its midpoint S, and .

Note that lines can bisect segments. However, lines cannot be bisected themselves because lines do not have a finite length.

Segment bisector m


Math Fact
In geometry, the definition of a segment bisector, or simply bisector, is “a segment, line, or ray that passes through the midpoint of a segment.” The definition says nothing about ending up with two congruent segments. It is the definition of a midpoint that tells us we end up with two congruent segments.

A midpoint on a segment will always give two relationships, the pair of congruent parts and the segment addition postulate, as seen in Figure below.

One or both relationships may be needed to solve a problem.


Segment relationships with a midpoint

Example 1
J is the midpoint of . GJ = (7x + 1) and JH = 3x + 9. Find GH.
Solution:


Example 2
 bisects   at point W. If PW = 2x + 7 and PQ = 3x + 24, find the length of PW.

Solution:

A special case of a bisector is the perpendicular bisector. This is a segment, line, or ray that passes through a segment’s midpoint at a right angle. Figure below shows perpendicular bisector  passing through midpoint E at a right angle.


Perpendicular bisector 

ANGLES IN POLYGONS


Key Ideas
There are relationships between the sides of a polygon with n sides and the measures of its interior and exterior angles. The sum of the exterior angles of a polygon always equals 360°. The sum of the interior angles of a polygon equals 180°(n − 2), where n equals the number of sides. A central angle of a regular polygon equals 360°/n, where n equals the number of sides.


Interior Angles of Polygons
An interior angle of a polygon is the interior angle formed by the intersection of two consecutive sides. Figure 2.11 shows the 4 interior angles of a quadrilateral.

Interior angles in a quadrilateral

A regular polygon is a polygon in which all the interior angles and all the sides are congruent. The sum of the interior angles of any polygon, and measure of a single interior angle of a regular polygon can be found using the following theorem.


Interior Angle Theorem for Polygons
The sum of the measures of the interior angles of a polygon with n sides is given by 180°(n − 2).
The measure of one interior angle of a regular polygon with n sides is given by .


Example 1
What is the measure of each interior angle of a regular hexagon?
Solution:
For a hexagon, n = 6. So each interior angle in a regular hexagon measures


Example 2
The sum of the measures of the interior angles of a polygon equals 540°. How many sides does the polygon have?
Solution:


Example 3
In regular hexagon ABCDEF, . Find m∠CBD.

Solution:



Math Tip
When faced with a complex problem, break it up into smaller pieces that you recognize. Then begin working your way toward your goal.


Exterior Angles in Polygons
An exterior angle of a polygon is formed by extending one side of the polygon. We consider only one extended side at each vertex, so a polygon with n sides has n exterior angles. Figure below illustrates the 5 exterior angles of a pentagon. Which side is extended does not matter because extending the second side at any vertex would result in a pair of vertical angles with equal measures.

Exterior angles in a pentagon
For regular polygons, the measure of one exterior angle equals , where n equals the number of sides.


Exterior Angle Theorem for Polygons
The sum of the measures of the exterior angles of a polygon with n sides equals 360°.
The measure of one exterior angle of a regular polygon with n sides equals .


Example
What is the measure of one exterior angle of a regular decagon?
Solution:
Each exterior angle measures  and n = 10.


Central Angles
A central angle is an angle with its vertex at the center of the polygon and rays through consecutive vertices, as shown in Figure 2.13. Central angles of a polygon are found in the same way as exterior angles.

Central angle ∠APB


Central Angles Formula
The sum of the central angles in a polygon is 360°.
The measure of each central angle in a regular polygon equals, where n is the number of sides.


Math Fact
If a regular polygon has an even number of sides, the center is the point of concurrency of segments with opposite vertices as their endpoints. If the polygon has an odd number of sides, the center is the point of concurrency of segments with a vertex at one endpoint and the midpoint of the opposite side as the other endpoint.


Example
Find the measures of angles x and y in a regular pentagon with center P, as shown in the accompanying figure.

Solution:
In a pentagon, n = 5. So each central angle measures

and y = 72°. Each interior angle of the pentagon measures
. m∠x is

the measure of an interior angle, or 54°.

PARALLEL LINES


Key Ideas
When two or more parallel lines are intersected by a transversal, any pair of angles formed are either congruent or supplementary. We can determine if two lines are parallel by looking at the measures of these pairs.

Two lines are parallel if and only if the following are true:
Alternate interior angles are congruent.
Same side interior angles are supplementary.
Corresponding angles are congruent.


Named Angle Pairs
Whenever two lines are intersected by another line, called a transversal, the eight angles shown in this Figure will be formed. Some of the angle pairs have names to make referencing them easier. Notice that the transversal divides the figure into two regions, on alternate sides of the transversal. In Figure 2.14, angles 1, 4, 5, and 8 are on one side. Angles 2, 3, 6, and 7 are on the other side. Next, the two lines intersected by the transversal form an interior region and an exterior region. Angles 3, 4, 5, and 6 are interior angles, while angles 1, 2, 7, and 8 are exterior angles.

The 8 angles formed when two lines are intersected by a transversal

Using these definitions, the named pairs are:
Alternate interior angles are ∠3 and ∠5 and also ∠4 and ∠6.
Same side interior angles are ∠4 and ∠5 and also ∠3 and ∠6.
Corresponding angles are ∠4 and ∠8, are ∠1 and ∠5, are ∠2 and ∠6, and also ∠3 and ∠7.
Alternate exterior angles are ∠1 and ∠7 and also ∠2 and ∠8.

Recognizing Angle Pairs
Alternate interior angles make a “Z” pattern that can help in recognizing them. The alternate interior angles are in the corners of the “Z,” as shown in the Figure. Same side interior angles make an “F” pattern. Those angles are in the corners of the “F” as shown in 2nd Figure.

Alternate interior angles in the corners of the “Z”

Same side interior angles in the corners of the “F”

Theorems Involving Parallel Lines and the Named Angle Pairs
When parallel lines are intersected by a transversal, a set of relationships is formed among the angles.


Corresponding Angle Postulate
Two lines intersected by a transversal are parallel if and only if the corresponding angles are congruent.

Alternate Interior Angle Theorem
Two lines intersected by a transversal are parallel if and only if the alternate interior angles are congruent.

Same Side Interior Angle Theorem
Two lines intersected by a transversal are parallel if and only if the same side interior angles are supplementary.

Alternate Exterior Angle Theorem
Two lines intersected by a transversal are parallel if and only if the alternate exterior interior angles are congruent.

All of the angle relationships involving parallel lines result in congruent pairs of angles or supplementary pairs of angles. Combining these three angle theorems with the linear pair and vertical angle postulates allows you to calculate the measure of all eight of the angles if the measure of just one angle is known.

Example 1
Line m || line n and m∠1 = 118°. Find the measures of the other angles.

Solution:

Example 2
Line m || line n. Find the value of x.

Solution: 
The indicated angles are supplementary, same side interior angles.


Example 3
Line m || line n. What is the value of x?

Solution:

alternate interior angles are congruent

Example 4
, m∠A = 32°, and m∠B = 38°. Find m∠ACD and m∠BCE.

Solution: 
Use the two pairs of alternate interior angles.


Using an Auxiliary Line
Sometimes we need to construct an additional line, called an auxiliary line, to help solve a parallel line problem.

Example 1
, m∠CDE = 38°, and m∠ABE = 32°. Find m∠BED.

Solution: 
We need to construct an auxiliary line,
, through point E and parallel to

and
.

congruent alternate interior angles congruent alternate interior angles angle addition

Example 2
Lines r and s are parallel. Find m∠1.

Solution: 
There are two “jogs” in the transversal. So we need two auxiliary lines, both parallel to lines r and s.

Determining if Two Lines Are Parallel
The relationships between corresponding, alternate interior, and same side interior angles are true when read in either direction. For example, the corresponding angle postulate tells us two things:

If two lines are parallel, then the corresponding angles are congruent.
If the corresponding angles are congruent, then the lines are parallel.

The second statement can be used to determine whether two lines are parallel by comparing the measures of any pairs of corresponding angles. The same can be done for alternate interior and same side interior angles.

Example 1
Lines m and n are intersected by transversal v, m∠4 = 52°, and m∠5 = 128°. Determine if lines m and n are parallel.

Solution:

Same side interior angles ∠4 and ∠7 are supplementary, so lines m and n are parallel.

Example 2
In quadrilateral ABCD, m∠A = 111°, m∠B = 69°, m∠C = 107°, and m∠D = 73°. How many pairs of parallel sides does ABCD have?
Solution:


because same side interior angles are supplementary.
m∠A + m∠B = 69° + 111° = 180°

and

are not parallel because the same side interior angles are not supplementary.



So quadrilateral ABCD has 1 pair of parallel sides.

Example 3
For what value of x is
?

Solution:
∠AEF and ∠CFH are corresponding angles. They must be congruent if
.


 

ANGLES AND SIDES IN TRIANGLES


Key Ideas
A set of theorems relate angle measures in triangles.
The sum of the measures of the interior angles of a triangle equals 180°.
The measure of any exterior angle of a triangle equals the sum of the measures of the nonadjacent interior angles.
If two sides of a triangle are congruent, then the angles opposite them are congruent.
All interior angles of an equilateral triangle measure 60°.


Triangle Angle Sum Theorem
The angle sum theorem for triangles is simply the polygon interior angle theorem evaluated for a polygon with three sides. The sum of the interior angles of a triangle equals 180°.


Math Fact
The triangle angle sum theorem can be used to help classify the type of triangle. The theorem is used to calculate the measures of each of the three angles in a triangle.


Example 1
The measures of the three angles in a triangle are (3x + 12)°, (4x − 6)°, and (2x + 12)°. Find the measure of each angle, and completely classify the type of triangle.
Solution: Set the sum of the angle measures equal to 180°.

The three angle measures are:

The triangle is isosceles because two angles are congruent, and acute because all angles measure less than 90°.

Example 2
In △LOV, m∠LOE = (5x − 68)°, m∠VOE = (2x + 7)°, m∠V = 28°, and

is an angle bisector. Find m∠L.

Solution:

By substituting x into the expressions for each angle, we find:

angle addition angle sum theorem in △LOV

Triangle Exterior Angle Theorem
The triangle exterior angle theorem relates the measure of an exterior angle of a triangle to the two nonadjacent interior angles. In Figure 2.17, m∠1 = m∠2 + m∠3.

Exterior angle 1 and nonadjacent angles 2 and 3
This theorem is particularly useful when exterior angles are expressed in terms of a variable.

Example 1
In △ABC, side  is extended through point A to point D. If m∠CAD = 140° and m∠ACB = 95°, find m∠ABC. Is △ABC isosceles?

Solution: From the exterior angle theorem,

	exterior angle theorem
	linear pair

△ABC is not isosceles since no two angles in the triangle are congruent.

Example 2
In △RST, m∠URT = (10x + 39)°, m∠RST = (7x + 4)°, and m∠STR = (6x + 8)°. Find m∠URT.

Solution: Apply the triangle exterior angle theorem,


Isosceles Triangles
An isosceles triangle has two congruent sides, and the angles across from those sides are always congruent. The congruent angles are called the base angles, and the congruent sides are called the legs. The side that differs in length is called the base, and the angle that differs in measure is called the vertex angle.

These parts are shown below.

Isosceles triangle


Isosceles Triangle Theorem
If two sides in a triangle are congruent, then the angles opposite those sides are congruent.


Converse of the Isosceles Triangle Theorem
If two angles in a triangle are congruent, then the sides opposite those angles are congruent.

If any one angle of an isosceles triangle is known, then the other angles can be determined.

Example 1
In this Figure, m∠C = 106°. Find m∠A and m∠B.
Solution: Base angles ∠A and ∠B are congruent.
Use the triangle angle sum theorem,


Example 2
In this figure, m∠A = 42°. Find m∠C.
Solution: Base angles ∠A and ∠B are congruent.


Example 3
In the accompanying figure,

and
. If m∠1 = 68°, what is m∠2?

Solution:

	triangle angle sum theorem

Example 4
In the accompanying figure,
, m∠ACD = (9x − 4)°, and m∠ABC = (5x − 10)°. Find m∠A.

Solution: △ABC is isosceles with ∠B ≅ ∠A. Use the exterior angle theorem,


Equilateral Triangles
An equilateral triangle is one in which all three sides are congruent and all three angles measure 60°.

Example
△ABC is equilateral, and D lies on
. m∠ADC = 102°. Find m∠BAD.

Solution:

	angles in an equilateral triangle measure 60°
	angle sum theorem in △ADC
	angle addition

Pythagorean Theorem
In a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse, as shown in Figure 2.19. The hypotenuse is always the longest leg of the triangle, which is opposite the 90° angle. Remember, the Pythagorean theorem applies only to right triangles.

Pythagorean theorem

In a right triangle with legs represented by lengths a and b and with hypotenuse c, a2 + b2 = c2.


Example 1
In △RST, ∠S measures 90°, RS = 6, and ST = 3. Find RT.
Solution: Since ∠S is the right angle, RT must be the hypotenuse.


Example 2
In the accompanying figure, m∠E = 45°, EO = GO, and EG = 8. Find EO and GO.

Solution:

	isosceles triangle theorem angle sum theorem △GEO is a right triangle, so Pythagorean theorem applies
	let EO = GO = x

Example 3
The length of a rectangle is 3 times its width. If the length of a diagonal is 10, what is the area of the rectangle?
Solution: The right triangle formed by the sides of the rectangle and a diagonal has legs equal to 3x and x and a hypotenuse equal to 10.


Example 4
In △ABC, m∠A = 90°, AC = 6, AB = 2x, and BC = x + 6. Find the area of the triangle.

Solution: