Bisect It

ID: sukin-jaris
Created by Illustrative MathIllustrative Math
Subject: Geometry
Grade: 9-12

Bisect It

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Problem 1

1) Select all quadrilaterals for which a diagonal is also a line of symmetry. Write each corresponding letter in the answer box and separate letters with commas.

a) trapezoid \quad\quad b) isosceles trapezoid \quad\quad c) parallelogram \quad\quad d) rhombus \quad\quad e) rectangle \quad\quad f) square

Problem 2

2) Show that diagonal EGEG is a line of symmetry for rhombus EFGHEFGH.

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Problem 3

3) ABDEABDE is an isosceles trapezoid. Priya makes a claim that triangle AEBAEB is congruent to triangle DBEDBE. Convince Priya this is not true.

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Problem 4

4) In quadrilateral ABCDABCD, triangle ADCADC is congruent to CBACBA. Show that ABCDABCD is a parallelogram.

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Problem 5

Priya is convinced the diagonals of the isosceles trapezoid are congruent. She knows that if she can prove triangles congruent that include the diagonals, then she will show that diagonals are also congruent. Help her complete the proof.

Given: ABCDABCD is an isosceles trapezoid.

Draw auxiliary lines that are diagonals 1 and 2 . ABAB is congruent to 3 because they are the same segment. We know angle BB and 4 are congruent. We know AEAE is congruent to 5 . Therefore, triangle ABEABE and 6 are congruent because of 7 . Finally, diagonal BEBE is congruent to 8 because 9 .

Fill in the blanks using items from the Bank of Terms below. Some items might be used more than once.

Bank of Terms: Side-Angle-Side Triangle Congruence Theorem, Corresponding parts of congruent figures are congruent, Triangle BADBAD, Angle AA, ADAD, BABA, BDBD, BEBE.

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5) Blanks 1 and 2, answers separated by a comma.

6) Blank 3

7) Blank 4

8) Blank 5

9) Blank 6

10) Blank 7

11) Blank 8

12) Blank 9

Problem 6

Given: AFAD\overline{AF} \cong \overline{AD}, FD\angle F \cong \angle D

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13) Is there enough information in the diagram to prove that triangle AFEAFE is congruent to triangle ADEADE?

True or false? Write below.

14) Explain your reasoning.

Problem 7

15) Triangle DACDAC is isosceles with congruent sides ADAD and ACAC. Which additional given information is sufficient for showing that triangle DBCDBC is isosceles? Select all that apply. Write each corresponding letter in the answer box and separate letters with commas.

a) Segment DBDB is congruent to segment BCBC. \quad\quad b) Segment ABAB is congruent to segment BDBD.

c) Angle ABDABD is congruent to angle ABCABC. \quad\quad d) Angle ADCADC is congruent to angle ACDACD.

e) ABAB is an angle bisector of DACDAC. \quad\quad f) Triangle BDABDA is congruent to triangle BDCBDC.

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