4. Tierra knows that right angles are congruent. To prove this she would need to use which important axiom below?

Transcription

1 Name: Date: The following set of exercises serves to review the important skills and ideas we have developed in this unit. Multiple Choice Practice suur 1. In the following diagram, it is known that ABC and EF be concluded from these givens? AB CB DBE CBE B is the midpoint of AC DBC is supplementary to DBA bisects AC at B. Which of the following cannot Line is a segment bisector, not necessarily an angle bisector. So, since we do not know if bisects, we cannot conclude that the two angles in are congruent. 2. Which of the following is not always true? vertical angles are congruent complements of congruent angles are congruent all right angles are congruent all obtuse angles are congruent All of these are true statements except choice. We can easily have two obtuse angles, say one that measures and one that measures, that are not congruent. 3. Alonzo knows that in the following diagram points A, B, C, and D are collinear and that AB BC BC CD. He correctly concludes that AB CD. Which of the following would be the correct reason for this conclusion? a midpoint divides a segment into two equal segments the whole is the sum of its parts the subtraction property of equality the addition property of equality 4. Tierra knows that right angles are congruent. To prove this she would need to use which important axiom below? equals may be substituted for equals when equals are added to equals the sums are equal when equals are subtracted from equals the differences are equal the whole is the sum of its parts He correctly used the subtraction property of equality. But subtracting the quantity from both sides of this equation, he produces the fact that. If and are both right angles then and. Thus and. This is the substitution property of equality.

2 5. In which of the following diagrams is there not enough information shown to prove the given congruence statement? ABC DBC ACB DEB Only choice does not have sufficient information as it results in a S.S.A. scenario which generally does not prove congruence. ABC DBC ABC DBC 6. In the following diagram it is given that AFC is congruent to DEB. Which of the following would justify the statement that AC is congruent to DB? corresponding parts of congruent triangles are congruent the side angle side theorem the addition property of equality the whole is the sum of its parts 7. In the diagram below, it is known that uuur uuur BD bisects ABC, AD is perpendicular to BA, and CD is perpendicular to uuur BC. Which of the following theorems could be used to prove that DAB is congruent to DCB? A.S.A. S.A.S. H.L. A.A.S. Because and are corresponding sides of congruent triangles we can conclude they are congruent by. We can mark up the diagram as shown based on the givens. So, the two triangles have two pairs of congruent angles. They also share a common side of, but this side is not included between the congruent angle pair. So, we use A.A.S.

3 8. Given that lines m and n shown below are parallel, which of the following pairs of numbered angles do not need to be congruent? 2 and 3 3 and 5 1 and 2 4 and 5 The pair of angles in choice are generally supplementary, not congruent. All other angle pairs are congruent. 9. In the diagram below, DE is parallel to GK and CA is perpendicular to BA. If m EAC 35 of the following is the measure of ABC? 35 o 45 o 55 o 145 o Because is parallel to,. Since the angles of a triangle must sum to we can then find that. o then which 10. Given the two lines m and n shown below cut by a transversal and the five numbered angles shown, which of the following pieces of information would not be enough to conclude that m and n are parallel? and 4 are supplementary and 4 are supplementary and will form a straight angle and so be supplementary even if the two lines aren't parallel. 11. In MNP it is given that NM is congruent to NP. Which of the following must be true? PNM PMN N lies on the perpendicular bisector of MP MPN MNP M lies on the perpendicular bisector of NP 12. In the diagram shown below, TR is perpendicular to RS, TW is perpendicular to WS, andtr is congruent to TW. Which of the following would be used to show that TRS TWS? S.S.S. S.A.S. A.A.S. H.L. Both triangles are right triangles, have a pair of congruent legs, and, and share a hypotenuse. So they are congruent by Hypotenuse-Leg.

4 Free Response Practice 13. In the following diagram, BD is perpendicular to AC and bisects ABC. Which triangle congruence theorem would be used to show that ADB is congruent to CDB? Justify your choice. You would use the Angle-Side-Angle (or ASA) Theorem show they are congruent. Since is perpendicular to both and are right angles and therefore congruent. Since bisects, is congruent to. Finally, the two triangles share side which is (included) between the two pairs of congruent angles. 14. In the diagram below, it is known that QUS is congruent to TUR. Prove that QUR is congruent to SUT. QUS TUR Given QUS QUR RUS TUR SUT RUS QUR RUS SUT RUS The whole is the sum of its parts. Substitution RUS RUS QUR SUT Reflexive property Subtraction property 15. In the diagram below, lines m and n are parallel and AB is congruent to AC. Prove that 1 is congruent to 2. Given and Alternate interior angles formed by parallel lines are congruent. Given Base angles of an isosceles triangle are congruent. Substitution property

5 16. In the diagram shown, HF bisects EHG and EH GH. Prove: EF GF bisects Given Given An angle bisectors divides an angle into two congruent angles. Given SAS Theorem CPCTC 17. In the diagram shown, NM and OQ are both congruent and parallel. Also, NR and OP are both perpendicular to MP. Prove: NR OP and are perpendicular to Given Corresponding angles created by parallel lines are congruent. Given and are right angles (7) (8) Perpendicular segments form right angles All right angles are congruent. Given (7) A.A.S. Theorem (8) CPCTC

6 18. In the diagram shown AB and CE intersect at D such that D is the midpoint of AB. BC is parallel to AE. Prove: D is also the midpoint of CE Given D is the midpoint of Alternate interior angles formed by parallel lines are congruent. Given (7) (8) D is the midpoint of A midpoint divides a segment into two congruent segments. Vertical angles are congruent. ASA Theorem * (7) CPCTC (8) Definition of a midpoint. * The two triangles can also be proven congruent using the AAS Theorem by first stating E C. 19. Serena needed to bisect A shown in the diagram below. She located points B and C such that AB is congruent to AC. She then located point D such that BD is congruent to CD and drew in uuur AD. Does uuur AD bisect BAC? Justify your response. Yes, does bisect. We know from how Serena did the construction that and. We also know that by the reflexive property. That means that is congruent to by the S.S.S. Theorem. Since the two triangles are congruent, based on CPCTC. This means that bisects by the definition of an angle bisector.

7 20. Given: ST VT and QT WT Prove: RT UT Given (7) (8) (9) Vertical angles are congruent. Given SAS Theorem CPCTC Given (already stated, but stating again for clarity) (7) Vertical angles are congruent. (8) ASA Theorem (9) CPCTC You can also prove that WTU QTR using the same line of reasoning and then use CPCTC to show that RT UT. 21. In the diagram shown, BD CD and BD bisects ABC. If o m ADB 70 as shown, then is DAB isosceles? Justify your response. No, is not isosceles. We can first calculate that because it is supplementary with. Then, since we know that. They both must measure so that the angles of sum to be. Since bisects then as well. Finally, we can use the fact that the angles of must sum to to find that. Since does not have two congruent angles it is not isosceles.

8 22. In quadrilateral EFGH shown below, both pairs of opposite sides are congruent. Prove: EF is parallel to HG (Hint: Draw in diagonal EG ) and Given Reflexive Property SSS Theorem CPCTC If a pair of alternate interior angles created by two lines crossed by a transversal are equal then the lines are parallel. 23. In the diagram shown, AEC is congruent to DEB, ABE is congruent to DCE and AE is congruent to DE. Prove that BEC is isosceles. Given (7) (8) (9) is isosceles The whole is the sum of its parts. Reflexive Property Subtraction Property Given Given (7) AAS Theorem (8) CPCTC (9) Definition of an isosceles triangle