UNIT 3 CIRCLES AND VOLUME Lesson 2: Inscribed Polygons and Circumscribed Triangles Instruction

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1 Prerequisite Skills This lesson requires the use of the following skills: finding measures of inscribed angles and/or their intercepted arcs understanding that an angle inscribed in a semicircle is a right angle calculating the supplement of an angle understanding the properties of special quadrilaterals Introduction One of the most famous drawings of all time is Leonardo da Vinci s Vitruvian Man. Da Vinci s sketch was of a man enclosed by a circle that touched the man s feet and hands. In this lesson, we will investigate the properties of quadrilaterals inscribed in a circle. U3-5

2 Key oncepts n inscribed quadrilateral is a quadrilateral whose vertices are on a circle. The opposite angles of an inscribed quadrilateral are supplementary. D + = 80 m + m = 80 m + m D= 80 Remember that the measure of an inscribed angle is half the measure of the intercepted arc. Rectangles and squares can always be inscribed within a circle. ommon Errors/Misconceptions mistakenly thinking that the diagonal of any inscribed quadrilateral is a diameter of the circle U3-6

3 Guided Practice Example onsider the inscribed quadrilateral in the following diagram. What are the relationships between the measures of the angles of an inscribed quadrilateral? D 52. Find the measure of. is an inscribed angle. Therefore, its measure will be equal to half the measure of the intercepted arc. The intercepted arc D has a measure of , or 74º. The measure of is 2 of 74, or 87º. U3-7

4 2. Find the measure of D. The intercepted arc has a measure of , or 86º. The measure of D is 2 of 86, or 93º. 3. What is the relationship between and D? Since the sum of the measures of and D equals 80º, and D are supplementary angles. 4. Does this same relationship exist between and? The intercepted arc D has a measure of , or 56º. The measure of is of 56, or 78º. 2 The intercepted arc D has a measure of , or 204º. The measure of is of 204, or 02º. 2 The sum of the measures of and also equals 80º; therefore, and are supplementary. 5. State your conclusion. The opposite angles of an inscribed quadrilateral are supplementary. U3-8

5 Example 2 onsider the inscribed quadrilateral below. Do the relationships discovered between the angles in Example still hold for the angles in this quadrilateral? E 74. alculate the measures of all four angles of quadrilateral E. intercepts E, so the measure of is half the measure of E. m = (04 74) 89 intercepts E, so the measure of is half the measure of E. m = (00 74) 87 intercepts E, so the measure of is half the measure of E. m = (00 82) 9 E intercepts, so the measure of E is half the measure of. m E = (04 82) 93 U3-9

6 2. Find the sum of the measures of and. The sum of the measures of and is equal to = State your conclusion. The measures of and sum to 80º, as do the measures of and E ; therefore, it is still true that opposite angles of an inscribed quadrilateral are supplementary. Example 3 Prove that the opposite angles of the given inscribed quadrilateral are supplementary. w z x D y. What is the sum of wº + xº + yº + zº? Together, the arcs create a circle that measures 360º; therefore, the sum of the arc measures is 360. U3-20

7 2. Find the measure of each angle of quadrilateral D. m = 2 ( x+ y) m = z) m = 2 ( x+ z) m D= x) 3. Find the sum of the measures of and. m + m = 2 ( x+ y) + z ) = 2 ( x+ y+ w+ z ) = (360) 80 2 = 4. Find the sum of the measures of and D. m + m D= 2 ( y+ z) + x) = 2 ( y+ z+ w+ z) = (360) = State your conclusion. m + m =80 and m + m D = 80. Therefore, each pair of opposite angles of an inscribed quadrilateral is supplementary. U3-2