Practicing Proofs

ID: bufuf-kusah
Created by Illustrative MathIllustrative Math
Subject: Geometry
Grade: 9-12

Practicing Proofs

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Problem 1


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1) Painters and carpenters use scaffolding to climb buildings from the outside. What shapes do you see? Why does one figure have more right angles?

Problem 2

2) Select all true statements based on the diagram. Write each corresponding letter in the answer box and separate letters with commas.

a) Angle CBECBE is congruent to angle ABEABE. \quad\quad b) Angle CEBCEB is congruent to angle DEADEA. \quad\quad c) Segment DADA is congruent to segment CBCB. \quad\quad d) Segment DCDC is congruent to segment ABAB. \quad\quad e) Line DCDC is parallel to line ABAB. \quad\quad f) Line DADA is parallel to line CBCB.

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Problem 3

3) Prove ABCDABCD is a parallelogram.

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Problem 4

4) Tyler has proven that triangle WYZWYZ is congruent to triangle WYXWYX using the Side-Side-Side Triangle Congruence Theorem. Why can he now conclude that diagonal WYWY bisects angles ZWXZWX and ZYXZYX?

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Problem 5

5) WXYZWXYZ is a kite. Angle WXYWXY has a measure of 133 degrees and angle ZYXZYX has a measure of 34 degrees. Find the measure of angle ZWYZWY.

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Problem 6

Elena is thinking through a proof using a reflection to show that the base angles of an isosceles triangle are congruent. Complete the missing information for her proof.

Call the midpoint of segment 1 . Construct the perpendicular bisector of segment CDCD. The perpendicular bisector of CDCD must go through BB since it's the midpoint. AA is also on the perpendicular of CDCD because the distance from AA to 2 is the same as the distance from AA to 3 . We want to show triangle ADCADC is congruent to triangle ACDACD. Reflect triangle ADCADC across line 4 . Since 5 is on the line of reflection, it definitely lines up with itself. DBDB is congruent to 6 since ABAB is the perpendicular bisector of CDCD. DD' will coincide with 7 since it is on the other side of a perpendicular line and the same distance from it (and that’s the definition of reflection!). CC' will coincide with 8 since it is on the other side of a perpendicular line and the same distance from it (and that’s the definition of reflection!). Since the rigid transformation will take triangle ADCADC onto triangle ACDACD, that means angle 9 will be taken onto angle ADCADC (they are corresponding parts under the same reflection), and therefore they are congruent.

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

Bank of Terms: AA, BB, CC, DD, ABAB, CBCB, ADCADC, ACDACD

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6) Blank 1

7) Blanks 2 and 3, answers separated by a comma.

8) Blank 4

9) Blank 5

10) Blank 6

11) Blank 7

12) Blank 8

13) Blanks 9 and 10, answers separated by a comma.

Problem 7

14) Segment EGEG is an angle bisector of angle FGHFGH. Noah wrote a proof to show that triangle HEGHEG is congruent to triangle FEGFEG. Noah's proof is not correct. Why is Noah's proof incorrect?

1. Side EGEG is congruent to side EGEG because they're the same segment.

2. Angle EGHEGH is congruent to angle EGFEGF because segment EGEG is an angle bisector of angle FGHFGH.

3. Angle HEGHEG is congruent to angle FEGFEG because segment EGEG is an angle bisector of angle FGHFGH.

4. By the Angle-Side-Angle Triangle Congruence Theorem, triangle HEGHEG is congruent to triangle FEGFEG.

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Problem 8

15) Figure HNMLKEFGHNMLKEFG is the image of figure ABCDKLMNABCDKLMN after being rotated 90 degrees counterclockwise around point KK. Draw an auxiliary line in figure ABCDKLMNABCDKLMN to create a quadrilateral. Draw the image of the auxiliary line when rotated 90 degrees counterclockwise around point KK.

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16) Write a congruence statement for the quadrilateral you created in figure ABCDKLMNABCDKLMN and the image of the quadrilateral in figure HNMLKEFGHNMLKEFG.