Chapter 2: Properties of Angles and Triangles
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1 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.
2 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: = 180
3 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: = 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: = 90 Converse: A statement that is formed by switching the premise and the conclusion of another statement.
4 Chapter 2: Properties of Angles and Triangles Section 2.1 Solving for Unknown Angles (a) (b) (c) (d) (e)
5 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.
6 Chapter 2: Properties of Angles and Triangles Section 2.1
7 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.
8 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.
9 Chapter 2: Properties of Angles and Triangles Section 2.1
10 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.
11 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)
12 Chapter 2: Properties of Angles and Triangles Section 2.2 (c) (d) (e)
13 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: Angle interior = 180 (n 2) n
14 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.
15 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 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.
16 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
17 Chapter 2: Properties of Angles and Triangles Section 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)
18 Chapter 2: Properties of Angles and Triangles Section 2.4 (b) (c)
19 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.
20 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 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.
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