UNIT 1 SIMILARITY, CONGRUENCE, AND PROOFS Lesson 5: Congruent Triangles Instruction
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1 Prerequisite Skills This lesson requires the use of the following skills: understanding that rigid motions maintain the shape and size of angles and segments, and that rigid motions include the transformations of reflections, rotations, and translations ability to identify corresponding pairs of sides and angles Introduction When a series of rigid motions is performed on a triangle, the result is a congruent triangle. When triangles are congruent, the corresponding parts of the triangles are also congruent. It is also true that if the corresponding parts of two triangles are congruent, then the triangles are congruent. It is possible to determine if triangles are congruent by measuring and comparing each angle and side, but this can take time. There is a set of congruence criteria that lets us determine whether triangles are congruent with less information. Key oncepts The criteria for triangle congruence, known as triangle congruence statements, provide the least amount of information needed to determine if two triangles are congruent. ach congruence statement refers to the corresponding parts of the triangles. y looking at the information about each triangle, you can determine whether the triangles are congruent. The Side-Side-Side (SSS) ongruence Statement states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. If it is known that the corresponding sides are congruent, it is understood that the corresponding angles are also congruent. The Side-ngle-Side (SS) ongruence Statement states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. U1-297
2 The included angle is the angle that is between the two congruent sides. Included angle Non-included angle is included between and. is included between and. is NOT included between and. is NOT included between and. The ngle-side-ngle ongruence Statement, or S, states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. The included side is the side that is between the two congruent angles. Included side Non-included side is included between and. is included between and. is NOT included between and. is NOT included between and. fourth congruence statement, angle-angle-side (S), states that if two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the triangles are congruent. This lesson will focus on the first three congruence statements: SSS, SS, and S. U1-298
3 The following diagram compares these three congruence statements. Side-Side-Side (SSS) Side-ngle-Side (SS) ngle-side-ngle (S) G X J T Q H Z Y W V S R XYZ TVW GHJ QRS ommon rrors/misconceptions misidentifying included sides and angles, resulting in the wrong congruence statement misreading congruency symbols of triangles changing the order of named triangles, causing parts to be incorrectly interpreted as congruent U1-299
4 Guided Practice xample 1 etermine which congruence statement, if any, can be used to show that PQR and STU are congruent. S P U T R Q 1. etermine which components of the triangles are congruent. ccording to the diagram, RP US, PQ ST, and TU QR. orresponding side lengths of the two triangles are identified as congruent. 2. etermine if this information is enough to state that all six corresponding parts of the two triangles are congruent. It is given that all side lengths of the two triangles are congruent; therefore, all their angles are also congruent. ecause all six corresponding parts of the two triangles are congruent, then the two triangles are congruent. 3. Summarize your findings. PQR STU because of the Side-Side-Side (SSS) ongruence Statement. U1-300
5 xample 2 etermine which congruence statement, if any, can be used to show that and are congruent. 1. etermine which components of the triangles are congruent. ccording to the diagram,,, and. Two corresponding side lengths of the two triangles and one corresponding angle are identified as congruent. 2. etermine if this information is enough to state that all six corresponding parts of the two triangles are congruent. Notice that the congruent angles are included angles, meaning the angles are between the sides that are marked as congruent. It is given that two sides and the included angle are congruent, so the two triangles are congruent. 3. Summarize your findings. because of the Side-ngle-Side (SS) ongruence Statement. U1-301
6 xample 3 etermine which congruence statement, if any, can be used to show that HIJ and KLM are congruent if HI KL, H K, and I L. 1. etermine which components of the triangles are congruent. One corresponding side length of the two triangles and two corresponding angles are identified as congruent. It is often helpful to draw a diagram of the triangles with the given information to see where the congruent side is in relation to the congruent angles. I H J K M L 2. etermine if this information is enough to state that all six corresponding parts of the two triangles are congruent. Notice that the congruent sides are included sides, meaning the sides are between the angles that are marked as congruent. It is given that the two angles and the included side are equivalent, so the two triangles are congruent. 3. Summarize your findings. HIJ KLM because of the ngle-side-ngle (S) ongruence Statement. U1-302
7 xample 4 etermine which congruence statement, if any, can be used to show that PQR and STU are congruent if PQ ST, PR SU, and Q T. 1. etermine which components of the triangles are equivalent. Two corresponding side lengths of the two triangles and one corresponding angle are identified as congruent. raw a diagram of the triangles with the given information to see where the congruent sides are in relation to the congruent angle. P Q R S T U U1-303
8 2. etermine if this information is enough to state that all six corresponding parts of the two triangles are congruent. Notice that the congruent angles are not included angles, meaning the angles are not between the sides that are marked as congruent. There is no congruence statement that allows us to state that the two triangles are congruent based on the given information. 3. Summarize your findings. It cannot be determined whether PQR and STU are congruent. U1-304
9 xample 5 etermine which congruence statement, if any, can be used to show that and are congruent. G 1. etermine which components of the triangles are congruent. Notice that the triangles overlap. If you have trouble seeing the two triangles, redraw each triangle. ccording to the diagram,,, and. One corresponding side length of the two triangles and two corresponding angles are identified as congruent. U1-305
10 2. etermine if this information is enough to state that all six corresponding parts of the two triangles are congruent. Notice that the congruent sides are included sides, meaning the sides are between the angles that are marked as congruent. It is given that two angles and the included side are equivalent, so the two triangles are congruent. 3. Summarize your findings. because of the ngle-side-ngle (S) ongruence Statement. U1-306
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