Geometric Terminology

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1 Geometric Terminology Across 3. An angle measuring Non coplanar, non intersecting lines. 6. Two angles that add to In a right triangle, one of the shorter sides. 9. Lines that form right angles. 10. An angle measuring less than Congruent angles formed by intersecting lines. 1. A polygon with all sides equal.. Longest side of a right triangle. Down 1. Triangle with at least two congruent sides.. An angle measuring Coplanar lines that never intersect. 5. Two angles that add to Equal. 7. Triangle with no equal sides. 13. An angle measuring more than Number of sides in a quadrilateral. Word List leg skew acute obtuse scalene isosceles equilateral hypotenuse perpendicular complementary supplementary congruent parallel straight vertical right four Milliken Publishing Company 1 MP4057

2 Angles of a Triangle The three angles in a triangle add to 180 : ( = 180 ). If each side is extended in one direction, exterior angles are formed: ( 4, 5, and 6). 5 An interior angle and its adjacent exterior angle are supplements: ( = 180 ) The measure of an exterior angle is equal to the sum of its remote interior angles: ( + 3 = 4). All three exterior angles add to 360 : ( = 360 ). Place the letter in the blank above its measure to reveal an interesting fact about our 49th state. E I H A C F G K W L O N S T R Milliken Publishing Company 4 MP4057 Y

3 Using Algebra with Complements and Supplements Remember: x = the angle 90 x = its complement 180 x = its supplement Set up an equation for each problem, then solve for x. Use your answer for x to find the angle measures for the problem. 1. The complement of an angle is five times the measure of the angle itself. Find the angle and its complement.. The supplement of an angle is 30 less than twice the measure of the angle itself. Find the angle and its supplement. 3. The supplement of an angle is twice as large as the angle itself. Find the angle and its supplement. 4. The complement of an angle is 6 less than twice the measure of the angle itself. Find the angle and its complement. 5. Three times the measure of the supplement of an angle is equal to eight times the measure of its complement. Find the angle, its complement, and its supplement. 6. Two angles are congruent and complementary. Find their measures. 7. Two angles are congruent and supplementary. Find their measures. 8. The complement of an angle is twice as large as the angle itself. Find the angle and its complement. 9. The complement of an angle is 10 less than the angle itself. Find the angle and its complement. 10. The supplement of an angle is 0 more than three times the angle itself. Find the angle and its supplement. Milliken Publishing Company 9 MP4057

4 The Pythagorean Theorem In a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse. the hypotenuse is opposite the right angle a c the legs form the right angle b Pythagorean Theorem: a + b = c Solve for the missing side. Use the decoder to find out what the numbers 3, 6, 10, and have in common A B E G H I L M N R S T U Y Milliken Publishing Company 14 MP4057

5 Congruent Triangles Three methods of proving triangles congruent: Side Side Side (SSS) 3 sides of one triangle are congruent to 3 sides of another triangle. Side Angle Side (SAS) sides and the included angle of one triangle are congruent to sides and the included angle of another triangle. Angle Side Angle (ASA) angles and the included side of one triangle are congruent to angles and the included side of another triangle. Use the diagrams and the information given to determine which of the above methods will prove the triangles congruent. Circle the letters beneath the correct method in the chart to reveal the mathematician who developed the symbol for congruence (@). A W B 1 3 X 4 5 C Y V Z L M N 16 P O 6 D 1. X is the midpoint of AD and BC. AB ^ BC ; DC ^ BC ; BX = XC 3. AB CD ; AB = CD 4. YW = YV; WZ = VZ 5. 7 = 9; 8 = WZ = VZ; YZ bisects WZV 7. AD and BC bisect each other 8. N is the midpoint of MO ; LM = PO; LN = PN 9. LM and PO are ^ to MO; 11 = 16; LM = PO 10. N is the midpoint of MO ; 1 = ; 13 = LM = PO; MN = NO; LN = PN 1. ZY bisects WYV; WY = YV = 6; X is the midpoint of AD 14. N is the midpoint of MO ; LN = PN; 13 = 14. YZ bisects WYV and WZV 16. DWYZ and DVYZ are equilateral SSS SAS ASA A G R C H O I M T T E D E S F T R H A I L E Z P L M D B P L E A S C I A L O B Q N P R M I Z O S Milliken Publishing Company 1 MP4057

6 Trapezoids All trapezoids have exactly one pair of parallel sides (called bases). An isosceles trapezoid has congruent legs, base angles, and diagonals. A right trapezoid has two right angles. Any trapezoid can be divided into a rectangle and triangle(s) by drawing altitudes between the bases. This will aid in finding measures of segments and angles. right trapezoid isosceles trapezoid general trapezoid Use the properties of trapezoids, rectangles, and right triangles to find the missing measures. Shade the answers below to find which U.S. state has borders that closely resemble a right trapezoid k 8 10 c 6 e 6 j 7 a b f g d h m r 1 w x p q t 0 u v n Milliken Publishing Company 7 MP4057

7 Volume of Regular Right Pyramids and Cones To find the volume of either of these solids, use this formula: Volume = 1 3 (Area of base) (altitude of the solid) Pyramids The base is a regular polygon; the altitude is the distance from the center of the base to the vertex (tip) of the pyramid. Cones The base is a circle; the altitude is the distance from the center of the circle to the vertex (tip) of the cone. Notice the right triangle formed in each of these solids. Use this triangle and the Pythagorean theorem to solve for needed measures when they are not given. altitude slant height 1 base edge or radius Find the volume of the solids described below. Use the decoder to reveal the name of the first woman to appear on a U.S. postage stamp. 1. Regular square pyramid with base edge 8 cm and altitude 1 cm.. Cone with radius 6 cm and altitude 8 cm. 3. Regular square pyramid with base edge 5 m and altitude 3 m. 4. Cone with radius 10 m and altitude 9 m. 5. Regular square pyramid with base edge 3 in. and altitude 5 in. 6. Cone with diameter 16 in. and altitude 6 in. 7. Regular square pyramid with base edge 1 cm and slant height 10 cm. 8. Cone with radius mm and slant height 30 mm. 9. Regular square pyramid with altitude 0 mm and slant height 5 mm. 10. Cone with altitude 10 ft. and slant height 6 ft. 11. Regular triangular pyramid with base edge 8 cm and altitude 1 cm. 190p ft 3 300p m 3 in mm 3 1p 3 mm 3 18p in cm cm 3 5 m 3 56 cm 3 96p cm 3 A G H I M N O R S T W Milliken Publishing Company 36 MP4057

8 Angles Formed by Chords, Secants, and Tangents Angles formed by chords: Add two intercepted arcs and divide by two. B A to find the measure of 1: D 1 x C AB + CD = = 70 Angles formed by secants and tangents: Subtract smaller intercepted arc from larger intercepted arc and divide by two. F K = FK GJ G J H L P M 3 3 = LP MP Remember: angles INSIDE : ADD arcs and divide by angles OUTSIDE : SUBTRACT arcs and divide by N Q T 4 R 4 = TQR TR S Use the diagrams above and the information given to find the missing measures. Use the decoder to reveal the basketball player who scored 100 points in a game on March, 196. A B C E H I L M N R T W Milliken Publishing Company 40 MP4057

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