EXERCISES Practice and Problem Solving
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1 XI ractice and roblem olving For more practice, see xtra ractice. ractice by xample lgebra Find the value of x in each parallelogram. xample (page xample (page 95 lgebra Find the value of x x - x + x + 5 5x + 6 Find the value of x and the length of each side. 0. 7x -. F 0 x + 6x + M x -.5 x - 0. x lgebra Find the value of a.... (6a (a - 0 (a + 50 (a + 0 Find the value of a and the measure of each angle in each parallelogram (a - a a 5a I (0a + 0 (7a + 8 K J xample (page 96 lgebra Find the values of x and y in $. 7. = x, = y +, = x +, = y 8. = x +, = y, = x, = y + 9. = y, = x +, = y, = x - 0. = x, = y +, = x, = y. = 8x, = 6y, = x +, = y Lesson 6- roperties of arallelograms 97-0
2 xample (page 97. Find and F in the figure at the right.. ewing uppose you don t have a ruler. xplain how to space four buttons equally on a shirt if you know where the first and last buttons must be placed and you have a large piece of lined paper. 8 8 F In the figure, the horizontal lines are parallel and = =. Find each length. W U. ZU 5. XZ.5 6. XU 7. Z Z Y 8. U 9. XV 0. YX. YV X. WX. WV V pply Your kills lgebra Use the given information to find the lengths of all four sides of $.. he perimeter is 8 in. is 5 in. less than. 5. he perimeter is 9 cm. is 7 cm more than twice. Need elp? Use the flow proof on p. 9. roof 6. eveloping roof omplete this paragraph proof of heorem 6- by filling in the blanks. iven: $ rove: > and > roof: is a parallelogram, therefore 6 a. 9 and 6 b. 9. & > & and & > &, because alternate interior angles are c. 9. > by the d. 9 roperty of ongruence. herefore # > # by e. 9.o, > and > because f hapter 6 uadrilaterals
3 eveloping roof xercises 7 and 8 ask you for two different flow proofs of heorem omplete this flow proof of heorem 6- by filling in the blanks. iven: $MN N rove: &NM > &N and &MN > &M MN is a. a. 9 M MN and N M b. 9 MN N MN N c. 9 N N M M e. 9 NM M NM M d. 9 MN N f. 9 NM M g. 9 eading Math For help with reading and solving xercise 8, see p. 0. NM N h Write a flow proof for heorem 6- that follows the plan on page 95. Find the measures of the numbered angles for each parallelogram MN M i rror nalysis rian states that V = 0 cm in the figure at the right. xplain why rian s statement may not be correct.. lgebra In a parallelogram one angle is 9 times the size of another. Find the measures of the angles. 5 cm V lgebra Find the value(s of the variable(s in each parallelogram.. 5. y 6. y 7 (z 5 (y 7. (y 6 8. x y 9. x (x x y y (x y 6 x 8 y Lesson 6- roperties of arallelograms 97-0
4 = x y + x - 5 x + 7 y + 9 x y x y x 8 y 5 x y 5. Writing xplain how to find the measures of the remaining three angles of a parallelogram if you already know the measure of one of the angles. 5. a. Open-nded ketch two parallelograms whose corresponding sides are congruent but whose corresponding angles are not congruent. b. ritical hinking Is there an congruence theorem for parallelograms? xplain. 55. eveloping roof proof of heorem 6- is outlined below. upply the reasons for each step. * * * F iven: 6 6 and > rove: > F raw lines through and parallel to * * * and intersecting at and F at. 5 6 F * * * a. 6 6 F and > b. and are parallelograms. c. > and > d. > e. 6 f. & > &, & > &, & > &5, and & > &6 g. & > &5 h. # > #F i. > F Need elp? For each of xercises 56 58, sketch the diagram and mark it as you think through a proof. roof Write a paragraph proof, a flow proof, or a two-column proof. 56. iven: $LN and $N rove: &L > & 57. iven: $LN and $N rove: L 6 N 58. iven: $LN and $N rove: & is supplementary to &. L hallenge roof Write a paragraph proof, a flow proof, or a two-column proof. 59. iven: $W and $XYZ rove: & > &X 60. iven: $W and $XYZ rove: XY 6 X Y Z W 6. iven: $ and bisects &. rove: bisects & hapter 6 uadrilaterals
5 roof 6. a. rove that if two sides and the included angle of one parallelogram are congruent to corresponding parts of another parallelogram, then the parallelograms are congruent. (int: rove that all the corresponding parts of the parallelograms are congruent. b. Is there a theorem similar to for trapezoids? xplain. tandardized est rep ridded esponse ake It to the N Online lesson quiz at Web ode: afa-060 Use the parallelogram at the right for xercises Find the indicated segment length or angle measure. 6. JM 6. ML 65. m&l 66. m&j M (7y he measures of three angles in a parallelogram measure 0, 60, and 0. Find the measure of the fourth angle. J x - K 9y x he measures of two angles in a parallelogram are and. Find the measure of one of the other two angles. 69. wo consecutive angles in a parallelogram have measures x + 5 and x - 0. Find the measure of the smaller angle. x L Mixed eview Lesson 6- etermine the most precise name for each figure Lesson What additional information do you need to prove # > # by the L heorem? * * Lesson - In the figure at the right, n. Find each measure. 7. m& 7. m& 75. m& 76. m& 9 Lesson 6- roperties of arallelograms 97-0
Objectives To use relationships among sides and angles of parallelograms To use relationships among diagonals of parallelograms
6-2 roperties of arallelograms ontent tandards.o.11 rove theorems about parallelogram. s include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each
EXERCISES Practice and Problem Solving
XI ractice and roblem olving or more practice, see xtra ractice. ractice by xample xample (page 224) In each diagram, the red and blue triangles are congruent. Identify their common side or angle.. K 2.
4-3. Triangle Congruence by ASA and AAS. Content Standard. Essential Understanding You can prove that two triangles are congruent
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Using Corresponding Parts of Congruent Triangles
4-4 Using orresponding arts of ongruent riangles ontent tandards G..5 Use congruence... criteria for triangles to solve problems and prove relationships in geometric figures. lso G..12 bjective o use triangle
Proving That a Quadrilateral Is a Parallelogram. To determine whether a quadrilateral is a parallelogram
- roving That a Quadrilateral Is a arallelogram ontent Standards G.O. rove theorems about parallelograms... the diagonals of a parallelogram bisect each other and its converse... lso G.ST. Objective To
Objectives To use the AA Postulate and the SAS and SSS Theorems To use similarity to find indirect measurements
7-3 roving riangles imilar ontent tandards G..5 Use... similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.G.5 rove the slope criteria for parallel and
Proof EXAMPLE EXAMPLE. Given:
4-7 hat ou ll earn o identify congruent overlapping triangles o prove two triangles congruent by first proving two other triangles congruent... nd hy o identify overlapping triangles in scaffolding, as
The hypotenuse and a leg of one triangle are congruent to the hypotenuse and a leg of the other triangle. THEOREM 5.2. right triangles, and
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Proving Congruence ASA, AAS
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Problem 2. Got It? Proving Triangle Parts Congruent to Measure Distance. Proof
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Lesson 13.1 The Premises of Geometry
Lesson 13.1 he remises of Geometry Name eriod ate 1. rovide the missing property of equality or arithmetic as a reason for each step to solve the equation. olve for x: 5(x 4) 15 2x 17 olution: 5(x 4) 15
Name Class Date. This shows that A corresponds to Q. Therefore, A Q. This shows that BC corresponds to RS. Therefore, BC RS.
Name lass ate eteaching ongruent igures iven, find corresponding parts using the names. rder matters. or example, or example, his shows that corresponds to. herefore,. his shows that corresponds to. herefore,.
5.3 Proving Triangles are
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Essential Question What are the properties of parallelograms?
7. roperties of arallelograms ssential uestion What are the properties of parallelograms? iscovering roperties of arallelograms Work with a partner. Use dynamic geometry software. a. onstruct any parallelogram
7.4 Showing Triangles are
7. howing riangles are imilar: and oal how that two triangles are similar using the and imilarity heorems. ey ords similar polygons p. he triangles in the avajo rug look similar. o show that they are similar,
Isosceles Triangles. leg. base
6 4 What ou ll Learn ou ll learn to identif and use properties of isosceles triangles. Isosceles riangles ecall from Lesson 5 that an isosceles triangle has at least two congruent sides. he congruent sides
To identify congruence transformations To prove triangle congruence using isometries
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To recognize congruent figures and their corresponding parts
4-1 ongruent igures ontent Standard Prepares for G.SR.5 Use congruence... criteria for triangles to solve problems and prove relationships in geometric figures. Objective o recognize congruent figures
To identify congruence transformations To prove triangle congruence using isometries
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9.4 Conditions for Rectangles, Rhombuses, and Squares
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Altitudes and Perpendicular Bisectors
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1. What is the sum of the measures of the angles in a triangle? Write the proof (Hint: it involves creating a parallel line.)
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Congruence and Similarity in Triangles. INVESTIGATE the Math. as shown in Colin s design. Explain how you know they are similar.
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To use and apply properties of isosceles and equilateral triangles
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To prove two triangles congruent using the SSS and SAS Postulates. Are the triangles below congruent? How do you know? 6 B 4
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7.5 Proportions and. Similar Triangles. Geo-Activity. Goal Use the Triangle Proportionality Theorem and its converse.
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