section 2.1: angle properties and parallel...

Chapter 2: Properties of Angles and Triangles Section 2.1

Chapter 2: Properties of Angles and Triangles

Section 2.1: Angle Properties and Parallel Lines

Terminology:

Transversal :

A line that intersects two or more other lines at distinct points.

Interior Angles:

Any angles formed by a transversal and two parallel lines that lie inside the

parallel lines.

Exterior Angles:

Any angles formed by a transversal and two parallel lines that lie outside the

parallel lines.


Corresponding Angles:

One interior angle and one exterior angle that are non-adjacent and on the same

side of the transversal. Corresponding angles are equal.

Alternate Interior Angles:

Interior angles on opposite sides of the transversal that are adjacent to different

parallel lines. Alternate interior angles are equal.

Alternate Exterior Angles:

Exterior angles on opposite sides of the transversal that are adjacent to different

parallel lines. Alternate interior angles are equal.

Co-Interior Angles:

Interior angles on the same side of the transversal are co-interior. Co-interior

angles have a sum of 180°. 𝑁𝑜𝑡𝑒: 105° + 75° = 180°


Transverse Angles:

Angles on opposite sides of the transversal and on opposite sides of the same

parallel line are transverse angles. Transverse angles are equal.

Supplementary Angles:

Angles on the same side of both a transversal and a parallel line are

supplementary. Supplementary angles have a sum of 180°.

𝑁𝑜𝑡𝑒: 108° + 72° = 180°

Complementary Angles:

When a right angle is split into multiple pieces, each angle involved is

complementary. Complementary Angles have a sum of 90°.

𝑁𝑜𝑡𝑒: 52° + 38° = 90°

Converse:

A statement that is formed by switching the premise and the conclusion of

another statement.


Solving for Unknown Angles

(a) (b)

(c) (d)

(e)


Determining Unknown Angles and Identifying Properties

For each situation below, determine the unknown angle and state the property that can

be used to find it.


Determining Parallel Lines

Two lines are parallel if the angles formed with the transversal and each line are equal.

Ex. In each situation determine if the set of lines are parallel. Justify your answer.


Angle Properties and Linear Equations

In each situation, use angle properties to create an equation and determine the

value of x.


Section 2.2: Angle Properties in Triangles

Terminology:

Exterior Angle: The angle that is formed by a side of a polygon and the extension of an adjacent side.

Non- Adjacent Interior Angles:

The two angles of a triangle that do not have the same vertex as an exterior angle.

∠𝐴 𝑎𝑛𝑑 ∠𝐵 𝑎𝑟𝑒 𝑛𝑜𝑛 − 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒𝑠 𝑡𝑜 𝑡ℎ𝑒 𝑒𝑥𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒 ∠𝐴𝐶𝐷.

Relationship Between Non-Adjacent Interior Angles and Exterior Angles

Determine a Relationship between the non-adjacent interior angles ∠A and ∠B and the

exterior angle ∠D.


Determining Unknown Angles in a Triangle

Ex. In each triangle, determine the measure of each labeled unknown angle.

(a)

(b)


(c)

(d)

(e)


Section 2.3: Angle Properties in Polygons

Terminology:

Convex Polygon: A Polygon in which each interior angle measures less than 180°. In a convex polygon, the

sum of the interior angles can be expressed as 𝑆(𝑛) = 180°(𝑛 − 2) where n is the number

of sides that the figure has and 𝑆 is the sum of the interior angles.

Sum of Interior Angles:

𝑆(𝑛) = 180°(𝑛 − 2)

Measure of Interior Angles:

𝐴𝑛𝑔𝑙𝑒𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 =180°(𝑛 − 2)

𝑛


Determining the Measure of Interior Angles in Regular Polygons

Ex. Determine the measure of the interior angles in each situation.

(a) Outdoor furniture and structures like gazebos sometimes use a regular hexagon

in their building plan. Determine the measure of each interior angle of a regular

hexagon.

(b) Determine the measure of each interior angle of a regular pentadecagon (a 15-

sided polygon).

Determining the Sum of Interior Angles Ex1. Determine the sum of the interior angles in a regular heptagon.

Ex2. Determine the sum of the interior angles in a regular hexadecagon.


Determine the Number of Sides Given Sum of Angles

Ex1. The sum of the measure of the interior angles of an unknown polygon is 3060°.

Determine the number of sides that the polygon has.

Ex2. Determine the number of sides of a given polygon if the sum of its interior angles

is 1440°.


Section 2.4: Proving Congruent Triangles

Terminology:

Congruence:

Two shapes are considered to be Congruent if they are exactly the same. That means that

the two shapes have exactly the same angles and exactly the same side lengths. The

symbol for congruence is ≅.

Conditions of Congruence in Triangles

There are three conditions that allow us to determine if two triangles are congruent:

1. Side-Side-Side (SSS):

If three pairs of corresponding sides are equal in lengths, then the triangles are

congruent. This is known as side-side-side congruence or SSS.

For example:

𝐴𝐵 = 𝑋𝑌

𝐵𝐶 = 𝑌𝑍

𝐴𝐶 = 𝑋𝑍

∴△ 𝐴𝐵𝐶 ≅△ 𝑋𝑌𝑍

2. Side-Angle-Side (SAS):

If two pairs of corresponding sides and the contained angles are equal, then the

triangles are congruent. This is known as side-angle-side congruence or SAS.

For example:

𝐴𝐵 = 𝑋𝑌

∠𝐵 = ∠𝑌

𝐵𝐶 = 𝑌𝑍



3. Angle-Side-Angle (ASA):

If two pairs of corresponding angles and the contained side are equal then the

triangles are congruent. This is known as angle-side-angle congruence or ASA.

For example:

∠𝐵 = ∠𝑌

𝐵𝐶 = 𝑌𝑍

∠𝐶 = ∠𝑍


Determining Congruence in Triangles

In each situation determine if each set of triangles is congruent. State which property

you used and what supports you have used.

(a)


(b)

(c)


Proving a Congruence Statement

In each situation, use the information to prove each congruence statement.

(a) Given that 𝐴𝐵 = 𝐶𝐷, Prove

that △ 𝐴𝐵𝐶 ≅△ 𝐶𝐷𝐸.

(b) Given that 𝑇𝑃 ⊥ 𝐴𝐶 and 𝐴𝑃 = 𝐶𝑃,

prove △ 𝑇𝐴𝐶 is isosceles.

(c) Given that 𝐴𝐸 and 𝐵𝐷 bisect each other at C

and that 𝐴𝐵 = 𝐸𝐷, prove ∠𝐴 = ∠𝐸.


(d) The main entrance to the Louve in France is through a large pyramid. The base of

each face is 35.42 m long. The other two edges of each face are equal length.

Prove the two base angles on each face are equal.

(e) Given that MD is the diameter of the circle

TA=TH. Prove ∠𝐴𝑀𝑇 = ∠𝐻𝑀𝑇.

(f) Given that 𝐴𝐵 = 𝐷𝐸 and ∠𝐴𝐵𝐶 = ∠𝐷𝐸𝐶.

Prove △ 𝐵𝐸𝐶 is isosceles.

section 2.1: angle properties and parallel...

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