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Chapter 2 Terms (some explanations and diagrams on the following pages) 2-1: Vertical Angles Adjacent Angles Adjacent angles around a point Linear Pair Supplementary Angles Complementary Angles 2-2 & 2-3: Median & Midpoint Angle Bisector Altitude Perpendicular Bisector 2-4: Corresponding Angles Alternate Interior Angles Alternate Exterior Angles Same Side Interior Angles The B-S Rule (a hint, not a reason) 2-5: Auxiliary Lines 2-6: Sum of the Angles in a Triangle Exterior Angle Isosceles Triangle Equilateral Triangle 2-7: Substitution Property Subtraction Property Formal Proofs - what should they look like? Keywords & what they do Vertical Angles: Create congruent angles Statement: Vertical angles create congruent angles <A = <D Then write an equation to it the question Supplementary angles: Add to 1800. Commonly found as a linear pair. Statement: Supplementary angles equal 1800 <A + <B = 1800 Then write an equation to it the question Substitution Property: If two things (angles, segments, numbers, etc) are equal to the same thing, they must be equal to each other. If <A = 1120 and <B = 1120, then <A = <B. <A 112 ° 112° <B Subtraction Property: Just like numbers or variable, the measure of an angle or a segment can be subtracted. A = 30° If <A + <B = <C, then <A = <C - <B B= 7 0° C=1 00° Keywords & what they do Triangle Keywords Medians: Create a midpoint; midpoints create two congruent segments Example: BD is a median Statement: AD ~ = DC Then write an equation to it the question Angle Bisectors: Create two congruent angles. Example: AD is an angle bisector Altitudes: Create right angles Example: BD is an altitude B A D C Statement: <BAD ~ = <CAD Then write an equation to it the question Statement: <BDA is a right angle or <BDC is a right angle Then write an equation to it the question Perpendicular bisectors: Create right angles AND create midpoints (which then create congruent segments) Ex: BD is a perpendicular bisector Statement: <BDA is a right angle or <BDC is a right angle ~ DC or AD = Then write an equation to it the question Keywords & what they do Parallel Lines cut by a Transversal Corresponding Angles: Match up in their position. They are congruent. Statement: Corresponding angles are congruent <A ~ = <E Then write an equation to it the example Alternate Interior Angles: On opposite (alternate) sides of the transversal, and inside (interior) of the parallel lines. They are congruent. Statement: Alternate Interior angles are congruent <C ~ = <F Then write an equation to it the example Alternate Exterior Angles: On opposite (alternate) sides of the transversal, and outside (exterior) of the parallel lines. They are congruent. Statement: Alternate Exterior angles are congruent <A ~ = <H Then write an equation to it the example Same Side Interior Angles: On the same side of the transversal, and inside (interior) of the parallel lines. They are supplementary Statement: Same Side Interior angles are supplementary <D + <F = 1800 Then write an equation to it the example Name: ______________________________________________________________ Date: _______________________ Period: _______ Directions: Answer the following questions completely on a separate sheet of paper. DO NOT COMPLETE THE WORK ON YOUR IPAD!!! IT MUST BE ON LOOSELEAF OR A NOTEBOOK!!! Make sure to show all work. Justify your calculations for all questions! 1) In ΞABC, BD is an altitude drawn to side AC. If the m < BDA = 5π₯ β 30, find the value of x. 2) In ΞWXY, XZ is the altitude drawn to side WY. Find the value of x if, m< Y = 2π₯ β 5 and m< ZXY = 2π₯ + 21. Round to the nearest tenth, if necessary. 3) In triangle PQR, QS is a median. If PS = 4π₯ β 2 and RS = 2π₯ + 4. Find the value of x. 4) In βππ π, segment QT is the bisector of < π ππ. If π < π ππ = 2π₯ + 17 and π < πππ = 12π₯ β 3, find the value of x. Find π < π ππ. 5) In βABC, BD is a perpendicular bisector. AD = 2y + 4 and CD = y + 12 and m<BDA = 5(x β 12). Find the value of x and y. Justify your calculations. 6) In triangle ABC, AD is the median drawn to side BC. If BC=150 and DC=3π₯ β 14, find the value of x. Round your answer to the nearest tenth. 7) βββββ π΄π΅ bisects <CAT. m<CAB = 2x β 2 and m<BAT = x + 6. Find m<CAT. βββββββ bisects right angle ABC. If the m<ABD is 6x β 3, then what is the value of x? 8) π΅π· 9) Point K is the midpoint of segment TJ. If the length of JK = 2x β 4 and the length of TK = x + 6. What is the value of x? What is the length of segment TJ? 10) Two angles are supplementary. One angle is 30Λ more than another angle. Find both angles. 11) Lines AB and CD intersect at E. If m<AEC = 6x β 27 and m<BEC = 8x + 11, find the measure of <BEC. 12) <J and <A are vertical angles. If the m<J = 3x+20 and the m<A = 5x-50, then what is the value of <J? 13) Lines AB and CD intersect at E. If m<AEC = x+30 and the m<DEB = 4x, then what is the value of <AED? Name: ______________________________________________________________ Date: _______________________ 14) In the diagram below, m<ABC = 165Λ. Find the value of x and m<CBD. Period: _______ β‘ββββββββ forms a straight angle. Using the diagram, find the m<DBC. Round to the 15) In the diagram below π΄π΅πΆ nearest tenth. (Not drawn to scale.) 16) Solve for x and y. 17) The measures of two supplementary angles are in the ratio 2:8. What is the measure of the larger angle? 18) <1 and <2 form a linear pair. The measure of <1 is 20 less than three times of <2. Find the measure of the larger angle. 19) Use the diagram to answer the questions below: a) Name an angle supplementary to < π»ππ½ and provide a reason for your calculation. b) If π < π»ππ½ = 38°, what is the measure of each of the following angles? Provide reasons for your calculations. ο§ < πΉππΊ ο§ < π»ππΊ ο§ < π΄ππ½ Name: ______________________________________________________________ Date: _______________________ 20) Find the measures of each labeled angle. Give a reason for your solution. 21) In the diagram below m||n. Using the diagram, find the value of x. 22) In the diagram below m||n. Using the diagram, find the m<A. 23) Find the following angles for each question: 24) Find π < π. 25) Find π < π. Period: _______ Name: ______________________________________________________________ 26) Find the value of π₯. Date: _______________________ Period: _______ 27) In ΞABC, m<A = x, m<B = 2x + 2, and m<C = 3x + 4. What is the value of x? 28) In the diagram below, ΞABC is shown with AC extended through point D. If m<BCD = 6x + 2, m<BAC = 3x + 15, and m<ABC = 2x β 1, what is the value of x? (1) 12 (2) 14 10 11 (3) 16 (4) 18 1 9 29) The measures of the angles of a triangle are represented by 5xβ7, 7x+6, and 4xβ11. Find the value of x. 30) Find the value of x in the diagram below. 31) In the figure on the right, Μ Μ Μ Μ π΄π΅ β₯ Μ Μ Μ Μ πΆπ· and Μ Μ Μ Μ π΅πΆ β₯ Μ Μ Μ Μ π·πΈ . Prove that < π΄π΅πΆ = π < πΆπ·πΈ. 32) In the diagram below, prove that the sum of the labeled angles is 180°.