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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.
Chapter 2: Properties of Angles and Triangles Section 2.1
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°
Chapter 2: Properties of Angles and Triangles Section 2.1
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.
Chapter 2: Properties of Angles and Triangles Section 2.1
Solving for Unknown Angles
(a) (b)
(c) (d)
(e)
Chapter 2: Properties of Angles and Triangles Section 2.1
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.
Chapter 2: Properties of Angles and Triangles Section 2.1
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.
Chapter 2: Properties of Angles and Triangles Section 2.1
Angle Properties and Linear Equations
In each situation, use angle properties to create an equation and determine the
value of x.
Chapter 2: Properties of Angles and Triangles Section 2.2
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.
Chapter 2: Properties of Angles and Triangles Section 2.2
Determining Unknown Angles in a Triangle
Ex. In each triangle, determine the measure of each labeled unknown angle.
(a)
(b)
Chapter 2: Properties of Angles and Triangles Section 2.3
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
Chapter 2: Properties of Angles and Triangles Section 2.3
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.
Chapter 2: Properties of Angles and Triangles Section 2.3
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°.
Chapter 2: Properties of Angles and Triangles Section 2.4
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
∴△ 𝐴𝐵𝐶 ≅△ 𝑋𝑌𝑍
Chapter 2: Properties of Angles and Triangles Section 2.4
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
∴△ 𝐴𝐵𝐶 ≅△ 𝑋𝑌𝑍
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)
Chapter 2: Properties of Angles and Triangles Section 2.4
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.
Chapter 2: Properties of Angles and Triangles Section 2.4
(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.