Geometry: Unit 3 Congruent Triangles Practice

Transcription

1 Notes 1. Read the information in the box below and add the definitions and notation to your tool kit. Then answer the questions below the box. Two figures are CONGRUENT if they have exactly the same size and shape. This means that one figure can be slid, turned, and/or flipped so that it fits exactly on the other figure. The symbol for congruent is. When two figures are congruent, we always list CORRESPONDING LABELS in the same order. For example, in the figures below, it is proper to say that ΔABC ΔDEF because A corresponds to D, B corresponds to E, and C corresponds to F. However, it would be incorrect to say ΔABC ΔDFE. We can show the corresponding parts easily with arrows as follows: The two triangles below are congruent. Which of the following statements are correctly written and which are not? Explain why and why not. Discuss your answers with your team. 2. Read the following information and record it in your tool kit. Then go on to the next problem. We know that a triangle has six parts: three sides and three angles. If two triangles are congruent, then the corresponding six parts are congruent, and vice versa.

2 3. DIRECTIONS: You have had the opportunity to see for yourself that when certain combinations of three parts of triangles are congruent, everyone must build the same triangle. This observation--and the conditions under which congruence is assured--will be useful when you compare the corresponding parts of two triangles to see if the two triangles are congruent. Read the conclusions about triangle congruence in the box below and add the properties to your tool kit. In order to conclude that two triangles are congruent (without showing that all six corresponding parts of the triangles are congruent), you need only show that certain combinations of three pairs of corresponding parts are congruent. These combinations, called TRIANGLE CONGRUENCE PROPERTIES, are: SSS This represents side--side--side, which means all three pairs of corresponding sides are congruent. SAS This represents side--angle--side, which means two pairs of corresponding sides AND the angle they form (the included angle) are congruent. ASA This represents angle--side--angle, which means two pairs of corresponding angles AND the side they share (the included side) are congruent. Once you have demonstrated that two triangles are congruent, you may state that any of the other pairs of corresponding parts are congruent.

3 Classwork 1. DIRECTIONS: Use the three triangle congruence properties and the information about each pair of triangles below to decide whether the two triangles are congruent. If they are, write a clear explanation justifying your response similar to the example below. Then write the correct sequence of vertices for the second triangle. If the triangles are not congruent, briefly explain why not. Discuss each problem with your teammates.

4 Homework 1. Assume that each picture shows a pair of congruent figures. Complete a correct congruence statement to illustrate which parts correspond. Remember: letter order is important! 2. Refer to the triangles below and the statements at right and determine which of the statements are correct and which are not. 3. SSA does not usually guarantee congruent triangles, EXCEPT when working with a right triangle. Read the box below and add the triangle congruence property to your tool kit. Then answer the question below the box. For pairs of right triangles, when one pair of corresponding legs are congruent and each hypotenuse is the same length, the two triangles are congruent by the HYPOTENUSE-LEG triangle congruence property, abbreviated HL. Explain why H (hypotenuse) and L (leg), rather than SSA, are the correct letters to use to abbreviate your conjecture in the previous problem.

5 4. Consider the two triangles at right. You have seen that if two angles and the included side of one triangle are equal to the corresponding side and angles of another, then the two triangles are congruent. We have labeled this relationship ASA. Notice that in ΔDUB, we have information about two angles and the included side (namely D, U, and DU), but in ΔWHS we have two angles and a side that is NOT included between the two known angles. a. If you knew that m W = 75, would the triangles be congruent? Why or why not? b. Calculate the measure of W. c. Do you think the order of corresponding congruent parts matters for two angles and a side? Explain. 5. Be sure that all of the congruence properties for triangles are listed in your tool kit with a marked diagram of a pair of triangles that illustrates each property. You should have: a. In which one(s) is the order important? SSS, SAS, ASA (AAS), and HL b. What other patterns have you seen that DO NOT work in all cases? Explain why they do not assure congruence between two triangles. 6. DIRECTIONS: Use the triangle congruence properties and the information about each pair of triangles below to decide whether the two triangles are congruent. If they are, explain how you know you can use one of the triangle congruence properties, then write its abbreviation and the correct sequence of vertices for the second triangle. If the triangles are not congruent, briefly explain why not.