B C. You try: What is the definition of an angle bisector?

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1 US Geometry 1 What is the definition of a midpoint? The midpoint of a line segment is the point that divides the segment into two congruent segments. That is, M is the midpoint of if M is on and M M. 1 enchmark 1 Study Guide What is the definition of an angle bisector? G.O O Find the measure of angle. m O m O m O O. Then classify the + (ngle dd. Post.) m O (Substitution Prop.) m O 5 (Subtraction Prop.) Find the measure of the angle. O O. Then classify Since 0 < m O < 90, then the angle is an acute angle. G.O.1 3 is the bisector of. If m 29, then m + m _?_ 3 is the bisector of. If m 4, then m + m _?_ is the bisector of, so m m 29. m + m m m 5 m m + m G.O.1 Page 1 of 17 M@WUS (US) 11/07/13

2 US Geometry enchmark 1 Study Guide 4 Points,, and are collinear, but they do not necessarily lie on the line in the order named. If 3and There are two possibilities:, what is? 4 Points,, and are collinear, but they do not necessarily lie on the line in the order named. OR If 9and 5, what is? G.O.1 5 Give an example of the Symmetric Property of ongruence. If, then. 5 Give an example of each of the following properties of congruence. a) Reflexive b) Transitive G.O.1 6 O 6 If O is the angle bisector of O, what kind of angles are O & O? (list as many angle pair relationships as you can) O and O are adjacent angles, a linear pair, and supplementary angles. O G.O.1 Page 2 of 17 M@WUS (US) 11/07/13

3 US Geometry enchmark 1 Study Guide 7 Given: m 43, is a complement of and is a supplement of, determine whether each statement is True or False. ) m 137 ) m 43 ) m 137 ) m 47 ) m 43 F) is a complement of m ) and ) are false., so is a supplement of, so m ) is true and ) and ) are false., 7 Given: m 27, is a complement of and is a supplement of, determine whether each statement is True or False. ) m 153 ) m 63 ) m 173 ) m 43 ) m + m 216, m m F) is false. G.O.1 If and are complementary, and m x, what is the m and m? m + m 90 x + m 90 m 90 x m + m + m 10 ( ) x + 90 x + m m 10 m 90 OR m + m + m m 10 m 90 If m x, what is the m and m? G.O.10 Page 3 of 17 M@WUS (US) 11/07/13

4 US Geometry enchmark 1 Study Guide 9 p q. Solve for x. 9 p q. Solve for x. ( 5x + 1 p ) ( 10x + 12) q The angles whose measurements are given are same-side interior or consecutive interior angles. ecause the lines are parallel, these angles are supplementary. Therefore, p ( 2x + 5) ( 5x 4) p q p (5x+ 1) + (10 x+ 12) 10 15x x 150 x 10 G.O.1 10 Given: 10 Given: Prove: Δ Δ Statements Given Reasons Prove: Δ Δ Reflexive Prop. of 3. Δ Δ 3. SSS Post G.SRT.5 Page 4 of 17 M@WUS (US) 11/07/13

5 US Geometry enchmark 1 Study Guide 11 onstruct a line through P parallel to line l. P l We will construct PW parallel to l using corresponding angles: raw any line through P and l. We will copy P using only a straightedge and compass. Make an arc from (using as center). Make the same arc from P: 11 o each of the following constructions: a) opy a line segment. b) isect a line segment. c) opy an angle. d) isect an angle. Y P X l P Measure how much opens up by putting the compass point and pencil point at X and Y. Make a mark to show that you measured correctly: Z Y P X Make that same mark from Z, crossing the arc you made from P: Z Y P W X raw PW. Y Z P W X PW will be parallel to line l because ZPW Y and are corresponding angles. G.O.12 Page 5 of 17 M@WUS (US) 11/07/13

6 US Geometry 12 Δ ΔF by which postulate or theorem? 12 enchmark 1 Study Guide Name the postulate or theorem by which the congruence statement is true. a) Δ Δ F F The S ongruence Postulate. F b) Δ Δ c) Δ Δ d) Δ Δ G.SRT.5 Page 6 of 17 M@WUS (US) 11/07/13

7 US Geometry enchmark 1 Study Guide etermine whether the following statements are True or False. ) Δ Δ ) Δ Δ ) Δ Δ ) Δ Δ ) y the lternate Interior ngle Theorem, and. Therefore, must correspond to and to. That happens in both ) and ) so they are true, while ) and ) are false. etermine whether the following statements are True or False. ) Δ Δ ) Δ Δ ) Δ Δ ) Δ Δ ) orresponding parts of congruent triangles are congruent. Using either of the true congruency statements in ) or ), would correspond to. Therefore, the two segments would be congruent and ) is true. G.SRT.5 14 x Find x Q 6 x T Δ ~ Δ by the SS Similarity Theorem 6 3 Therefore, and vertical angles are congruent x 2 3 2x 9 x 9 2, G.SRT.5 N Find x. L Page 7 of 17 M@WUS (US) 11/07/13

8 US Geometry enchmark 1 Study Guide 15 The sides of a pentagon have lengths 2, 3,, 9, and 10. The longest side of a similar pentagon is 20. Find the perimeter of the second pentagon. omparing the lengths of the longest side of each pentagon: the larger pentagon is twice as large as the smaller pentagon. Therefore the perimeter of the larger pentagon is twice the perimeter of the smaller pentagon. 15 The sides of a hexagon have lengths 5, 6, 10, 12, 15, and 21. The longest side of a similar hexagon is 14. Find the perimeter of the second hexagon. OR Find the scale factor of the similar polygons smaller pentagon 10 larger pentagon Let x 1 the length of the smallest side. 2 1 x 2 1 x 4 1 then 3 1 x 2 2 x 6 2 Similarly, x 3 16 and x 4 1. The perimeter of the second pentagon is units. OR The ratio of the perimeters of similar polygons is equal to the scale factor: x x x x x 64 The perimeter of the second pentagon is 64 units. G.SRT.2 Page of 17 M@WUS (US) 11/07/13

9 US Geometry enchmark 1 Study Guide 16 omplete the statement: x x 16 a) Write the similarity statement for the triangles that are similar. 10 If, then Justify your reasoning x? x ecause, (corresponding s are if the lines are parallel) and for the same reason,. y the Similarity postulate, Δ : Δ : 10 + x b) How do you know the triangles are similar? c) Solve for x x 7 G.SRT.5 17 Given: Δ ~ Δ ZYX List all the angles that are congruent from the similarity statement and then write the statement of proportionality for the corresponding sides. Z Y X ZY YX ZX 17 Given: QRST ~ GFH List all the angles that are congruent from the similarity statement and then write the statement of proportionality for the corresponding sides. G.SRT.2 Page 9 of 17 M@WUS (US) 11/07/13

10 US Geometry enchmark 1 Study Guide 1 W X 130 Y Z ( 12x + 20) 1 7x y y 4x Find the value of x and y and justify your reasoning. If two parallel lines are cut by a transversal, then same side interior angles are supplementary. Therefore, m Y + m ZY 10. Substituting the expressions in the diagram we can solve for x: 4x + ( 12x+ 20) 10 16x x 160 x 10 ( 4x + 9) 72 Find the value of x and y and justify your reasoning. WX and YX are a linear pair so they are supplementary angles which means m WX + m YX 10. Substituting 130 for m WX, we can solve for m YX : m YX 10 m YX 50 Similarly XY and ZY are a linear pair so they are supplementary angles which means m XY m ZY 10 12x + 20 for +. Substituting ( ) m ZY and then 10 for x, we can solve for m XY : m XY+ 12x ( ) ( ) m XY m XY m XY 40 Finally, the 3 interior angles of a triangle add up to 10 so m YX + m XY + m XY 10 and through substitution we can set up an equation to solve for y: 50 + y y y 90 G.O.1 Page 10 of 17 M@WUS (US) 11/07/13

11 US Geometry enchmark 1 Study Guide 19 List all the possible ways you can prove the triangles below are congruent. 19 List all the possible ways you can prove the triangles below are congruent. SSS ongruence Postulate SS ongruence Postulate G.SRT.5 20 H 20 K K G F P Q H N F M etermine whether the following statements are True or False. ), G, and are collinear. ) G,, and K are coplanar. ) FG and G are a linear pair. ) G and K are parallel lines. ) G K etermine whether the following statements are True or False. ),, and F are collinear ) K is in plane M ) F and K are a linear pair. ) H and F are parallel lines. ) H and are coplanar, G, and, cannot lie on one line so they are not collinear. G,, and K lie in plane Q so they are coplanar. FG and G are a linear pair. G and K are skew lines, not parallel. G K (Vertical angles are congruent) ), ), and ) are True. G.O.1 Page 11 of 17 M@WUS (US) 11/07/13

12 US Geometry enchmark 1 Study Guide 21 L n 21 J 72 K M N etermine whether each of the following statements must be True based on the given diagram above. ) ΔNKJ Δ LKM ) ΔNKJ ~ Δ MKL ) n 72 ) LM JN ) LM JN etermine whether each of the following statements must be True based on the given diagram above. ) Δ ~ Δ ) Δ Δ ) Δ Δ ) ) KN KM (ef. of segments) JKN MKL (Vertical s are ) ΔNJK Δ MLK (SS Postulate) Keeping the corresponding parts in the correct order we can rewrite this congruency statement: ΔNKJ Δ MKL This means ) is false. It also means ) is true. ongruent figures are also similar with a scale factor of 1. From the congruency statement J L, so n 72, and ) is true. From the congruency statement LM JN, so ) is true. We cannot determine whether LM JN, so ) is not necessarily true. Therefore ), ), and ) must be true. G.SRT.5 nd of Study Guide Page 12 of 17 M@WUS (US) 11/07/13

13 US Geometry enchmark 1 Study Guide 1 2 You Try Solutions: n angle bisector is a ray that is equidistant from the sides of an angle, OR n angle bisector is a ray that cuts an angle into two congruent angles. m O + m O m O m O 122 m O (ngle dd. Post.) (Substitution Prop.) (Simplifying) Since 90 < m O < 10, then the angle is an obtuse angle. 4 5 There are two possibilities: (nswers may vary.) a. b. If and or or , then. 3 m + m? is the bisector of m m 6 O and O are adjacent angles, congruent angles, and complementary angles. m + m m m + m 4 2m 4 m 24 7 is a complement of m ) is false and ) is true., so m + m is a supplement of, so m ) and ) are false. m + m ) is true. Page 13 of 17 M@WUS (US) 11/07/13

14 US Geometry enchmark 1 Study Guide 9 m + m + m 10 x m 10 x + m 90 m 90 x m + m 10 ( x ) m m x 90 m 90 + x The angles whose measurements are given are alternate interior angles. ecause the lines are parallel, these angles are congruent. Therefore, 2x+ 5 5x 4 9 3x 3 x 11 a. Given To copy, first draw a ray or line and pick a point to call : Measure with a compass by putting the point on one end and drawing an arc on the other end. Make that same arc from : will ' ' be congruent to b. Given To bisect, make an arc above and below with the compass at (using as center) and opened 1 more than halfway (with radius > 2 m): 10 Statements Δ Δ Reasons 1. Given 2. Vertical angles are congruent 3. S ongruence Postulate o the same thing using as center, using the same radius as the previous step. The intersections of the arcs will be on the perpendicular bisector. raw a line segment, ray, or line that connects the two points of intersection, and you have bisected : bisects. ontinued on next page Page 14 of 17 M@WUS (US) 11/07/13

15 US Geometry 11 c. Given: First draw a ray or line and pick a point to call : Then draw an arc using as the center and draw that same arc from : enchmark 1 Study Guide a. HL ongruence Theorem b. SSS ongruence Postulate c. SS ongruence Postulate d. S ongruence Postulate Measure with your compass and use that same setting to make a mark from : (Reflexive Prop.) and is a right angle so the triangles are congruent by the HL Theorem. Since and, then Δ Δ. Therefore, ) is true and ), ), and ) are false. Since corresponds to in the congruency statement, then by PT. ) is true as well. ` raw a ray from through the intersection of the two arcs, and will be congruent to : 14 ΔQT ~ Δ NL by the Similarity d. Given 6 10 Make an arc from : Using the same radius (compass opening), make arcs from the points of intersection, and : F will bisect : Q x Q QT N NL 6 x x 4. x T N L F Page 15 of 17 M@WUS (US) 11/07/13

16 US Geometry The scale factor of the second hexagon to the first is perimeter of 2nd x 3 perimeter of 1st x x ( ) 2gg x x The perimeter of the second hexagon is 46 units a) Δ : Δ b) Δ : Δ by the SS Similarity Theorem 10 c) x x x 2x 1 x and enchmark 1 Study Guide If two parallel lines are cut by a transversal, then alternate interior angles are congruent. Therefore, and m 7x. The 3 interior angles of a triangle add up to 10 so m + m + m 10 and through substitution we can set up an equation to solve for x: 4x x ( ) 11x SS ongruence Postulate S ongruence Postulate S ongruence Theorem 11x 99 x 9 If two parallel lines are cut by a transversal, then alternate interior angles are congruent. Therefore, and ( 4x+ 9) y. Substituting 9 for x, we can solve for y. 4x+ 9 y ( ) ( ) y 45 y 17 QRST ~ GFH Q G R F S H T QR RS ST QT GF FH H G Page 16 of 17 M@WUS (US) 11/07/13

17 US Geometry enchmark 1 Study Guide 20,, and F are collinear because they lie on the same line. K is not in plane M but intersects it at point. F and K are not a linear pair, because the non-shared sides aren t opposite rays. H and F are not parallel lines. If you extended the lines they would intersect. H and are coplanar since they are both in plane M Therefore only ) and ) are true. 21 (Reflexive prop. of ) Δ Δ (SSS Postulate) Keeping the corresponding parts in the correct order we can rewrite this congruency statement: Δ Δ This means ) is false since the congruent parts don t correspond. We can also rewrite the congruency statement this way: Δ Δ This means ) is false and ) is true. From any of the congruency statements above, so ) is true. Note that for angles only the vertex of the angle matters in the congruency statement. From any of the congruency statements above. If two lines are cut by a transversal and alternate interior angles are congruent, then the two lines are parallel so and ) is true. Page 17 of 17 M@WUS (US) 11/07/13