Example 1. Find the angle measure of angle b, using opposite angles.

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Jodie Maxwell
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1 2..1 Exploring Parallel Lines Vertically opposite angles are equal When two lines intersect, the opposite angles are equal. Supplementary angles add to 180 Two (or more) adjacent angles on the same side of a line add to 180. Complementary angles add to 90 Two (or more) adjacent angles in a right angle add to 90. Example 1. Find the angle measure of angle b, using opposite angles. 20

2 Example 2. Find the angle measure of angle b, using supplementary angles. Example 3. Find the angle measure of angle b, using complementary angles. Try. Find the angle measure of angle b. 21

3 A transversal is a line that intersects two or more other lines at distinct points. Explore Parallel Lines Example 4. Draw a transversal that crosses two parallel lines (below) at an angle other than 90. Label every angle formed between intersecting lines with a unique lower-casee letter, and measure each angle in degrees. 22

4 When a transversal intersects not equal. a pair of non-parallel lines, the corresponding angles are Example 5. In each diagram, determine whether AB is parallel to CD.. Explain how you know. 23

5 2.2 Angles Formed by Parallel Lines Any pair of parallel lines makes an F-shape with a transversal that crosses them. The angles marked a are equal and are called corresponding angles. Explore Corresponding Angles Example 1. On the diagrams below mark three other pairs of F-shaped corresponding angles and label them b, c, and d. (Note that the F-shape can be backwards or upside down). Example 2. Determine the measures of the angles marked by letters. 24

6 Any pair of parallel lines makes a Z-shape with a transversal that crosses them. The angles marked a are equal and are called alternate interior angles. Explore Alternate Interior Angles Example 3. On the diagrams below mark two pairs of Z-shaped alternate interior angles and label them b and c. (Note that the Z-shape can be backwards). Example 4. Determine the measures of the angles marked by letters. 25

7 Any pair of parallel lines makes a C-shape with a transversal that crosses them. The angles marked o and x are called co-interior angles. They are NOT equal. They add to 180 Explore Co-Interior Angles Example 5. On the diagrams below mark two pairs of C-shaped co-interior angles and label them o and x. (Note that the C-shape can be backwards). Example 6. Calculate the measures of the three remaining angles 48 in the parallelogram. 26

8 Use Reasoning to Determine Unknown Angles Example 7. Determine the measures of a, b, c, and d. Use Angle Properties to Prove that Lines Are Parallel Example 8. One side of a cell phone tower will be built as shown. Use the angle measures to prove that braces CG, BF, and AE are parallel. 27

9 2.3 Angles Properties in Triangles In any triangle, the sum of the measures of the interior angles is proven to be 180. Use Angle Sums to Determine Angle Measures Example 1. Determine the measure of each indicated angle. Use Reasoning to Determine the Relationship between the Exterior and Interior Angles of a Triangle Example 2. Determine the relationship between an exterior angle of a triangle and its non-adjacent interior angles. 28

10 Try. Prove The measure of any exterior angle of a triangle is proven to be equal to the sum of the measures of the two non-adjacent interior angles. Use Exterior Angle to Determine Angle Measure Example 3. Determine the measure of each indicated angle. 29

11 Try. Determine the measure of each indicated angle. Use Reasoning to Determine Solve Problems Example 4. Determine the measures of, and. 30

12 2.4 Angles Properties in Polygons Convex Polygon A convex polygon is any polygon in which each interior angle measures less than 180. Concave Polygon A concave polygon is a polygon in which one or more interior angle measures more than 180. Determine Convex and Concave Polygons Example 1. State if each polygon is concave or convex. Try. State if each polygon is concave or convex. 31

13 Determine Properties of Angles in Polygons Example 2. Use the following table to determine how to find the sum of the interior angles of a polygon from the number of sides. For a polygon with n sides, it takes a minimum of (n 2) triangle(s) to cover it. Therefore, the sum of the interior angles is 180 (n 2). Try. Calculate the sum of the measures of the interior angles of a polygon with 9 sides. Try. Calculate the sum of the measures of the interior angles of a polygon with the given sides. a) 12 b) 15 32

14 Example 3. Determine the number of sides of a polygon whose interior angle sum equals 4140 Try. Determine the number of sides of a polygon whose interior angle sum equals: a) 720 b) 1260 c) 2880 For a regular polygon with n sides, the measure of each interior angle is Example 4. A regular polygon is a polygon with all sides equal and all angles equal. How can you find the measure of each interior angles of a regular pentagon? Try. What is the measure of each interior angle in a regular decagon (10-sided)? 33

15 Example 5. Determine the number of sides of a regular polygon whose interior angle measures 170. Try. Determine the number of sides of a regular polygon whose interior angle measures; a) 156 b) 175 c) 171 An exterior angle is formed between a ray formed by extending the sides of polygon in one direction and the next side of the polygon (adjacent to the ray). Example 6. Determine the measure of each exterior angle of a regular pentagon? The exterior angle of an n-sided regular polygon measures 360 The sum of the measures of the exterior angles of an n-sided regular polygon is 360 Try. Determine the number of sides of a regular polygon whose exterior angle measures a) 15 b) 40 c) 1 34

Unit 8 Chapter 3 Properties of Angles and Triangles

Unit 8 Chapter 3 Properties of Angles and Triangles May 16 7:01 PM Types of lines 1) Parallel Lines lines that do not (and will not) cross each other are labeled using matching arrowheads are always the