Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

 A B C D Homework

Solutions for Session 4: Homework

See solutions for Problems: H1 | H2 | H3 | H4

 Problem H1 Reasoning may vary. One way to find the measures of the angles is as follows: m5 is 45°, since it is a vertical angle to the given 45° angle. Since angles inside a triangle add up to 180°, m7 must be 70°. This means that m6 is 110° because m6 and m7 add up to 180°. Similarly, m8 must be 115°. m11 is 110°, since it is vertical to 6. m12 is 70°, m10 is 115°, and m9 is 65°. Since 12 and 3 are corresponding, m3 is 70°. Similarly, since 9 and 4 are corresponding, m4 must be 65°. Finally, using vertical angles, m1 is 65°, and m2 is 70°.

Problem H2

 a. m1, m2, and m3 add up to 180° because they lie on a straight line. b. 1 is the same as 5 because they are corresponding angles. c. 2 is the same as 6 because they are vertical angles. d. 3 is the same as 4 since they are corresponding angles. e. Since m1, m2, and m3 add up to 180°, and because they respectively equal 5, 6, and 4, it follows that m4, m5, and m6 add up to 180°. Since the reasoning that led us to the conclusion did not use any specific angle measure in the given picture, it holds in general. Note that this is a proof that shows there are 180° in every triangle!

Problem H3

 a. The measure is 90° (or 360° / 4). b. The measure is 180° (or 360° / 2). c. The measure is 120° (or 360° / 3). d. If a central angle cuts off an arc of one n-th of the full circle, its measure is 360° / n.

Problem H4

 a. The central angle is larger. b. It is twice as large. c. Answers will vary. One way to answer the question is to use the software to sketch the figure, then measure the two angles. If you tested this for several cases, it would lead to a conjecture that can be made here: Angles inscribed in circles have measures that are half the measure of the arc they intercept (cut off), which is equivalent to the measure of the central angle. Try it.