geometry a - unit 4: triangles and parallelograms lessons 16-19

examine parallelogram abcd. sides cd and ab have lengths of y + 18 and 4y, respectively. determine the value of y and answer the following question. what is the length of ab?

24

lesson 16

more triangle congruence

use the information and diagram to complete the problem. given: ab ∥ed bc∥df bc ≅df prove: △abc ≅ △edf match each statement in the proof with the correct reason.

0.67 of 1 1. ab ∥ed bc∥df bc ≅df : given 2. ∠bca ≅∠dfe, ∠bac ≅∠def : alternative interior angles theorem 3. △abc ≅ △edf : hl congruence theorem

use the figure and the information to complete the proof. given: abcd is a parallelogram. prove: the diagonals ac and bd bisect each other. match each statement in the proof with the correct reason.

0.67 of 1 1. abcd is a parallelogram : given 2. ab ≅ cd, ad ≅ bc : corresponding parts of congruent triangles are congruent 3. 9 ≅ 11 : vertical angles theorem 4. ab cd : definition of parallelogram 5. 2 ≅ 5 : alternative interior angles 6. aeb ≅ ced : asa congruence theorem 7. ae ≅ ce and be ≅ de : opposite sides of a parallelogram are congruent 8. e is the midpoint of ac, e is the midpoint of bd : definition of midpoint 9. ac and bd bisect each other : definition of bisect

use the figure and the information to complete the proof. given: abcd is a parallelogram. prove: ∠a is supplementary to ∠d. match each statement in the proof with the correct reason.

1. abcd is a parallelogram : given 2. ab cd : definition of parallelogram 3. ad is a diagonal of ab and cd : definition of diagonal 4. ∠a and ∠d are same-side interior angles : definition of same-side interior angles 5. ∠a is supplementary to ∠d : same-side interior angles theorem

examine parallelogram abcd. angles a and d have measures of x and (x+30), respectively. what is the value of x?

75

examine the following diagram and information. given: ad≅ bc, ∠cad ≅∠acb prove: quadrilateral abcd is a parallelogram. you can prove that quadrilateral abcd is a parallelogram by showing that ad∥bc because if one pair of opposite sides are both parallel and congruent, the quadrilateral is a parallelogram. which plan should you use to prove ad∥bc?

angles cad and acb are alternate interior angles, and they are congruent. so, ad∥bc by the converse of the alternate interior angles theorem.

use the triangles to answer the question. if △abc ≅ △fde, what is the measure of ∠acb?

34.7

lesson 17

properties of parallelograms

lesson 18

prove a figure is a parallelogram

lina wants to place a triangular design element into a table she is building. what additional information is needed for lina to use the hl congruence theorem to prove that △ghj ≅△ihj?

gj ≅ ij

use the diagram and information to complete the proof. gven: abdc is a quadrilateral. ab∥cd, ∠a ≅∠d prove: abdc is a parallelogram. match each statement in the proof with the correct reason.

1. abdc is a quadrilateral. ab∥cd, ∠a ≅∠d : given 2. cb ≅ cb : reflexive property of congruence 3. ∠abc ≅ ∠dcb : alternative interior angles theorem 4. abc ≅ dcb : aas congruence theorem 5. ab ≅ cd : corresponding parts of congruent triangles are congruent 6. abdc is a parallelogram : one pair of opposite sides is both parallel and congruent

use the following diagram and information to answer the question. given: abcdis a parallelogram. ad ≅ ad ove wants to prove that abcd is also a rhombus. he uses the properties of parallelograms to show ad ≅ cd and ad ≅cb. what can ove conclude from this to prove that abcd is a rhombus?

ove can conclude that ab ≅ cd ≅ad ≅ cb due to the substitution property of equality, and thus abcd is a rhombus.

lesson 19

special parallelograms

use the triangles to answer the question. if △abc ≅ △def, what is the length of df?

13

examine parallelogram abcd. ∠A = 8x + 2 and ∠B = 9x + 8 what is the value of x?

10

examine the quadrilateral. which of the following properties can be used to prove quadrilateral wxyz is a parallelogram? select all that apply.

wx ∥ yz and wz ∥ xy wx ∥ yz and wx ≅ yz wx ≅ yz and wz ≅xy

use the diagram and information to complete the proof. given: m∠s + m∠t = 180, m∠t + m∠q = 180 prove: quadrilateral qrst is a parallelogram. match each numbered statement in the proof with the correct reason.

1. given 2. definition of supplementary angles 3. if an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.

use the information in the diagram to prove △hij ≅△lkj match each statement in the proof with the correct reason.

1. given : ∠hij ≅∠lkj hi ≅kl 2. vertical angles are congruent : ∠hji ≅ ∠ljk 3. aas congruence theorem : △hij ≅△lkj

use the information and diagram to complete the problem. given: j is the midpoint of hl △hij and △lkj are right triangles. ij ≅kj prove: △hij ≅△lkj match each statement in the proof with the correct reason.

0.5 of 1 1. j is the midpoint of hl △hij and △lkj are right triangles. ij ≅kj : given 2. j bisects hl : definition of bisect 3. hj ≅ lj : definition of midpoint 4. △hij ≅△lkj : hl congruence theorem

examine parallelogram abcd and carefully read the description of paola's proof. given: ab∥cd and ad∥bc and ac is a diagonal of both sets of parallel lines. paola proves that ∠b ≅∠d. she first uses alternate interior angles theorem, then she finds ac ≅ ac by the reflexive property. next, paola uses the asa congruence theorem to show △abc ≅△cda. since corresponding parts of congruent triangles are congruent, ∠b ≅∠d. which pairs of angles could paola have concluded are congruent using the alternate interior angles theorem?

∠1 ≅ ∠2 and ∠3 ≅∠4

jordan welds a triangular gusset to strengthen the frame of her recreational vehicle. she compares it to another gusset, as shown in the following figures. what additional measurements must jordan know to use the aas congruence theorem to prove that the two gussets are congruent?

∠abc and ∠def measure 87 and ∠acb and ∠dfe measure 35

examine parallelogram wxyz and the following proof, which is missing statement 3. given: segment xz is a diagonal for wz and xy. prove: wz ≅ xy, xw ≅ yz which answer correctly fills in the blank for statement 3 of the proof to show that wz ≅ xy, xw ≅ yz.

∠wxz ≅∠yzx and ∠wzx ≅∠yxz

use the diagram and information to determine steps 4 through 9 of the proof. given: ∠5 ≅∠6, lo∥mn prove: quadrilateral lmno is a parallelogram. match each numbered statement for steps 4 through 9 with the correct reason.

4. given 5. aas congruence theorem 6. corresponding parts of congruent triangles are congruent 7. angle addition postulate 8. substitution property of equality 9. if both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

use the following diagram and information to answer the question. given: lmno is a parallelogram. △lmn ≅△onm ari wants to prove that lmno is a square. he begins by using properties of parallelograms and congruent triangles to prove that all sides of lmno are congruent. he then shows that ∠m ≅∠n because they are corresponding parts of congruent triangles. knowing that consecutive angles in a parallelogram are supplementary allows him to show that m∠m = m∠n = 90. how can ari complete the proof? select all that apply.

ari can use the definition of parallelogram to state ∠l ≅∠n and ∠m ≅∠o. then he can use the definition of congruence and substitution to show that m∠l = m∠m = m∠n = m∠o = 90. thus, parallelogram lmno is a square by the definition of a square. ari can state that lmno is a rectangle because it has at least one right angle. since rectangles have four right angles, he can state that lmno is a square by the definition of a square.

use the diagram and information to answer the question. given: de∥gf m∠D + m∠E = 180 prove: quadrilateral degf is a parallelogram. which answer correctly describes how to complete the proof?

show that ∠d and ∠e are supplementary by definition. then, since ∠d is supplementary to both of its consecutive angles, quadrilateral degf is a parallelogram.

use the diagram and information to answer the question. given: v is the midpoint of xz, ∠1 ≅∠2 prove: quadrilateral wxyz is a parallelogram. you can prove that klmn is a parallelogram by showing that its diagonals bisect each other. which plan should you use to prove that the diagonals bisect each other?

use ∠1 ≅∠2, ∠3 ≅∠4, and xv ≅zv to show △wvz ≅△yvx by the aas congruence theorem. then, use wv ≅ yv and xv ≅ zv to prove wy and xz bisect each other.

use the diagram and information to answer the question. let k be the midpoint of gi and hj. is there enough information to prove quadrilateral ghij is a parallelogram?

yes, since k is a midpoint for the segments, then k bisects gi and hj. if the diagonals of a quadrilateral bisect each other, then quadrilateral ghij is a parallelogram.

use the diagram and information to determine steps 4 through 8 of the proof. given: ∠dae ≅∠bce, e is the midpoint of ac. prove: quadrilateral abcd is a parallelogram. match each numbered statement for steps 4 through 8 with the correct reason.

4. vertical angles theorem 5. asa congruence theorem 6. corresponding parts of congruent triangles are congruent 7. definition of bisect 8. if diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

examine parallelogram abcd. segments ce and be have lengths 15 − x and 2x, respectively. what is the value of x?

5

use the diagram and information to determine steps 3 through 7 of the proof. given: ∠t ≅∠r, qr∥ts prove: quadrilateral qrst is a parallelogram. match each numbered statement for steps 3 through 7 with the correct reason.

1. alternative interior angles theorem 2. reflexive property of congruence 3. aas congruence theorem 4. corresponding parts of congruent triangles are congruent 5. if one pair of opposite sides are both parallel and congruent, then the quadrilateral is a parallelogram.

use the diagram and the information to complete the proof. given: △wxz and △yxz are right triangles wx ≅yz prove: ∠yzx≅∠wzx match each statement in the proof with the correct reason.

1. △wxz and △yxz are right triangles wx ≅yz : given 2. wz≅ wz : reflexive property of congruence 3. wxz ≅ yxz : hl congruence theorem 4. ∠yzx≅∠wzx : corresponding parts of congruent angles are congruent