Warm up Exercise. 1. Find the missing angle measure in the figures below: Lesson 54 Angle Relationships & Lesson 55 Nets.notebook.
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1 Warm up Exercise 1. Find the missing angle measure in the figures below: 45 o 29 o 30 o a d 49 o b c 1
2 Lesson 54: Angle Relationships Intersecting lines form pairs of adjacent angles and pairs of opposite angles. Adjacent Angles: 1 and 3 3 and 4 4 and 2 2 and Opposite Angles: 1 and 4 2 and 3 Adjacent angles share a common vertex and a common side but do not overlap. Opposite angles are formed by two intersecting lines and share the same vertex but do not share a side. Opposite angles are also called vertical angles. Vertical angles are congruent. 2
3 Lesson 54 Angle Relationships & Lesson 55 Nets.notebook Angles formed by two intersecting lines are related. If we know the measure of one of the angles, then we can find the measure of the other angles Together 1 and 2 form a straight angle measuring 180 degrees. If 1 measures 120 degrees, then 2 measures 60 degrees. Likewise, 1 and 3 form a straight angle, so 3 also measures 60 degrees and vertical 2 and 3 both measure 60 degrees. 3
4 Since 3 and 4 form a straight angle (as do 2 and 4), we find that m 4=120 o. Thus, vertical angles 1 and 4 both measure to be 120 o. 120 o 60 o 60 o 120 o Two angles whose measures total 180 o are called supplementary angles. We say that 2 is the supplement of 1 and that 1 is the supplement of 2. Supplementary angles may be adjacent angles, like 1 and 2, but it is not necessary that they be adjacent angles. 4
5 Angle Pairs Formed by Two Intersecting Lines Adjacent angles are supplementary. Opposite angles (verticle angles) are congruent. Example: Refer to this figure to find the measure of angles 1, 2, 3, and o 5
6 Notice that the measure of angles 2 and 3 in the last example total 90 o. Two angles whose measures total 90 o are complementary angles, so 3 is the complement of 2 is the complement of 3. Angles Paired by Combined Measures Supplementary: Two angles totally 180 o Complementary: Two angles totally 90 o In the following figures we name several pairs of angles formed by parallel lines cut by a third line called a transversal. If the transversal were perpendicular to the parallel lines, then all angles formed would be right angles. The transversal shown on the next slide is not perpendicular, so there are four obtuse angles that are the same measure and four acute angles that are the same measure. We will provide justification for these conclusions in a later lesson. 6
7 Corresponding angles are on the same side of the transversal and on the same side of each of the parallel lines. Corresponding angles of parallel lines are congruent. One pair of corresponding angles is 1 and 5. Name three more pairs of corresponding angles Alternate sides of Transversal Alternate interior angles are on opposite sides of the transversal and between the parallel lines. Alternate interior angles of parallel lines are congruent. One pair of alternate interior angles is 3 and 6. Name another pair of alternate interior angles. Alternate exterior angles are on opposite sides of the transversal and outside of the parallel lines. Alternate exterior angles of parallel lines are congruent. One pair of alternate exterior angles is 1 and 8. Nam another pair of alternate exterior angles.
8 Congruent Angle Pairs Formed by a Transversal Cutting Two Parallel Lines Corresponding angles: Same side of transversal, same side of lines Alternate interior angles: Opposite side of transversal, between lines Alternate exterior angles: Opposite sides of transversal, outside of lines Example: In the figure parallel lines p and q are cut by a transversal t. The measure of 8 is 55 o. What are the measures of the other angles shown? t p q 8
9 Example: Structural engineers design triangles into buildings, bridges, and towers because triangles are rigid. The sides of triangles do not shift when force is applied like the sides of quadrilaterals do. A railroad bridge is built with a steel truss in strengthened by triangles that keep the bridge straight under the weight of a train. A B C D E 40 O F G H I J Knowing one acute angle in this truss is sufficient to find the measures of all the angles. Find the measures of angles ABF, FBC, and FGB. Segments that look parallel are parallel, and segments that look perpendicular are perpendicular. 9
10 Lesson 55: Nets of Prisms, Cylinders, Pyramids, and Cones If we think of the surface of a solid as a hollow cardboard shell, then cutting open and spreading out the cardboard creates a net of the solid. For example, here we show a net for a pyramid with a square base. Example: This net represents the surfaces of what geometric solid? Sketch the solid and describe how this surface area formula relates to each part of the net: S=2 rh + 2 r 2 10
11 Example: Sketch a net for this triangular prism. 11
12 Example: Sketch the front, top, and right side views of this figure. Then sketch a net for this figure. Describe how the two sets of sketches are related. 3 cm 4 cm 2 cm 12
13 Homework: Lesson 54: a j & Lesson 55: a d 1 13
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