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Triangle Congruence Key Ideas When proving two triangles are congruent, we do not need to prove all three pairs of sides and all three pairs of angles congruent. There are five shortcuts commonly used—the SAS, SSS, ASA, AAS, and HL criteria. Each criterion requires only three pairs of parts to be congruent for us to conclude the triangles are congruent.
The Triangle Congruence Criterion The SAS Criterion Figure shows two triangles with two pairs of corresponding sides congruent and the pair of included angles congruent. The included angle is the angle whose vertex is the shared endpoint of the two congruent sides. Having only these three pairs of corresponding parts congruent is sufficient to prove the triangles are congruent. We call this the SAS criterion. Triangles with SAS congruence SAS Criterion Two triangles are congruent if two pairs of corresponding sides and the pair of included angles are congruent. It is important to determine where the angle lies in relation to the sides when using SAS. If the angle is not the included angle, then we do not have SAS and the triangles may not be congruent. Having two sides and the nonincluded angle allows a motion that is not rigid. Figure 6.2 shows how a triangle with two sides congruent and the nonincluded angle congruent could move in a nonrigid manner. Side can pivot out, with side extended to complete the triangle. can extend because its length was not specified, and ∠B can change since it was not specified either. The new configuration is now obtuse, whereas the original figure was acute. A nonrigid triangle with parts angle-side-side congruent Example 1 Which of the following pairs of triangles can be proven congruent using the SAS criterion? Solution: Choice (3) is the only choice in which both triangles have a pair of sides and the included angle labeled congruent. A congruence statement between a pair of triangles will specify the pairs of corresponding parts through the order in which the vertices are listed. Example 2 Write a congruence statement for the following pair of triangles. Solution: The triangles are congruent by the SAS criterion. The congruence statement ΔABC ≅ ΔTRS will match corresponding parts in the correct order. Often additional information is needed beyond sides and angles explicitly marked on the figure. Vertical angles, shared sides, midpoints, and parallel lines show up frequently. Example 3 Can the two triangles be proven to be congruent? If so, state which criterion is used. Write a congruence statement. Solution: The triangles are congruent by SAS, with congruent to itself by the reflexive property as the second pair of sides. The congruence statement is ΔABC ≅ ΔDBC. Example 4 What one other piece of information must be provided before one can conclude that the two triangles are congruent by SAS? (1) ∠AEB ≅ ∠CED (2) E is the midpoint of (3) E is the midpoint of (4)
Solution: (3) The midpoint would allow you to conclude that . Along with the vertical angles ∠AEB ≅ ∠CED and the marked sides , there is enough information to conclude the triangles are congruent by SAS. Example 5 For the pair of triangles shown, demonstrate the use of the SAS criteria by doing the following. (A) Write an appropriate congruence statement. (B) For each of the corresponding pairs of parts used to establish congruence, give a justification for why those parts are congruent. Given: E is the midpoint of Solution: ΔABE ≅ CDE is given information. ∠A ≅ ∠C because alternate interior angles formed by parallel lines are congruent. because a midpoint divides a segment into two congruent segments. The SSS, ASA, and AAS Criterion Besides SAS, the criteria SSS, ASA, and AAS are sufficient to prove two triangles congruent. As with SAS, the relative position of the sides and angles matters for ASA and AAS. In ASA, the side must be the included side; its endpoints must be the vertices of the two angles. In AAS, the side is not the included side. Figure 6.3 illustrates the SSS, ASA, and AAS criterion. Triangles congruent by the SSS, ASA, and AAS criterion Math Fact Most combinations of 3 angles and sides are sufficient to prove two triangles congruent. The two exceptions are ASS and AAA. As mentioned in the previous section, ASS allows for two different triangles to be formed from the specified parts using a nonrigid motion. AAA is not a congruent criterion because it allows for multiple triangles to be formed from the specified parts as well, actually an infinite number of possible triangles. As we will see in Chapter 8, AAA is one of the similarity criteria. Any triangle with the same shape will have all three angles congruent. However, the triangles could be enlargements or reductions of one another. Example 1 For each pair of triangles, determine if they can be proven congruent. If so, state the criterion. Solution: AAS SSS ASA SAS You can apply any applicable angle or segment relationships to help show that corresponding parts of triangles are congruent. Example 2 For each pair of triangles, identify which congruence criterion could be used to prove the triangles congruent. Explain why each of the pairs of parts used are congruent.
Solution: SAS: ∠BCA ≅ ∠DCE by vertical angles. and because C is a midpoint that forms two pairs of congruent segments.
ASA: ∠QST ≅ ∠RST because an angle bisector forms two congruent angles. by the reflexive property. ∠QTS ≅ ∠RTS because perpendicular lines form congruent right angles.
AAS: Two pairs of congruent angles are given. The included side is congruent to itself by the reflexive property. SSS: Two pairs of congruent sides are given. The third side is congruent to itself by the reflexive property. Justification of the Triangle Congruence Criterion The SAS criterion can be proven using rigid motions. Assuming the only three given congruent parts, we can show that there exists a composition of rigid motions that will map ΔDEF onto ΔABC, shown in Figure 6.4a. First translate the ΔDEF by vector so that the vertices D and A (the vertices with the congruent angles) coincide as shown in Figure. Justification of the SAS criterion for triangle congruence
Next, rotate about point D by ∠FDC, as shown in 6.4c, so that coincides . We know this mapping is possible because . Finally, reflect ΔDEF over so that E maps to B as shown in 6.4d. Point E must map to B because . ΔDEF is mapped to ΔABC, and the triangles are congruent.
The justification for the ASA criterion is similar to that for the SAS criterion. The justification for the SSS criterion differs. Given ΔABC and ΔDEF with and as shown in Figure 6.5a, we can translate ΔDEF such that points D and A coincide.
Then rotate so that coincides with as shown in Figure 6.5b. Because we have no given congruent angles, the final part of the justification is somewhat different than for SAS and ASA. Construct diagonal as shown in Figure c. ΔEAB is isosceles, making ∠DEB ≅ ∠DBE. ΔECB is also isosceles, making ∠CEB ≅ ∠CBE. The SAS criterion now applies and the triangles are congruent. Justification of the SSS criterion for triangle congruence The Hypotenuse-Leg Criterion The hypotenuse-leg (HL) criterion applies only to right triangles. When given two right triangles, if a pair of legs are congruent and the two hypotenuses are congruent, then the triangles are congruent by HL. Figure 6.6 shows two triangles congruent by HL. The HL criterion for triangle congruence The HL criterion can be justified by applying the Pythagorean theorem and showing that the remaining legs in each triangle are congruent to each other. Since the two right angles are congruent, the triangles must be congruent by the SAS criterion.
Proving Triangles Congruent Key Ideas Two triangles can be proven to be congruent using the congruence postulates—SSS, SAS, ASA, AAS, and HL. Each of the required pairs of corresponding parts must be shown to be congruent before the triangles can be stated to be congruent. Using the Congruence Postulates We can write a formal proof showing that two triangles are congruent using one of the 5 congruence postulates. Given information and information inferred from the figure are used to show that each of the two or three corresponding pairs of parts are congruent. A good strategy will help you write a good proof. The following steps will help you consistently write a good proof. Know the material—Be familiar with the congruence postulates and the “tools” of geometry. The tools include vocabulary as well as angle and segment relationships. The better you understand the tools of geometry, the easier you will find writing a proof. It all starts with the figure—Sketch a figure if no figure is provided. Then use the given information and anything you can infer from the information to mark your figure with any relevant congruent markings, parallel markings, and right angle marking. These marks are crucial to helping you determine which congruence postulate to use. Plan a strategy—Determine which congruence postulate to use. The markings should help. If you do not have enough corresponding parts marked congruent, you need to think about how you can use the given information to conclude other parts are congruent.
Look for the following:
Vertical angles Shared sides and angles Linear pairs Supplementary and complementary angles Parallel lines and perpendicular lines Midpoints and bisectors of segments Angle bisectors Altitudes, medians, and perpendicular bisectors in triangles Isosceles triangle theorem, exterior angle theorem, and triangle angle sum theorem If you are missing parts—If you are still missing congruent parts, try focusing on the part or parts that would complete the postulate. Be careful not to come up with an invalid justification just because it would let you complete the proof! Write your statements and reasons—Remember these few important rules as you write your statements and reasons: Statements are specific to the proof. This is where you state facts about named points, lines, segments, angles, triangles, and so on. Reasons never mention a point, line, segment, angle, triangle, and so forth. Reasons can only be given as postulates, theorems, and definitions. If you cannot come up with a good reason, do not use the statement. The order sometimes matters. If one statement depends on another statement, the dependent statement should come after. How to tell when you are done—You must have one statement in your proof showing congruence for each of the corresponding pairs of parts specified by the congruence postulate. Remember that you cannot finish with the triangle congruence statement until each part of the postulate is proven. One strategy is to label with an (S) or (A) each line that proves a pair of required sides or angles congruent. So if you are using SAS, there should be two lines labeled (S) and one line labeled (A). Another strategy is to write out your postulate and make checkboxes under each part. Check off the boxes as each required line is written.
As a final check, keep in mind that a good triangle congruence proof should have: All required statements to demonstrate congruence No statements that cannot be justified No statement that does not help achieve the desired proof It is not incorrect to include an unnecessary statement as long as the statement is true. However, unnecessary lines in your proof will only provide an opportunity for error. Example 1 Given: bisects at E, ∠A ≅ ∠D Prove: ΔABE ≅ ΔDCE Solution: Mark the figure with the corresponding parts we know are congruent. The given bisector lets us conclude that . From congruent vertical angles, we know ∠AEB ≅ ∠DEC. Finally, we have the given congruent pair ∠A ≅ ∠D. From the marked figure, we see that the ASA postulate applies. When writing the proof, we state that E is a midpoint before concluding that sides and are congruent.
Math Fact Not all correct proofs are equal. Writing proofs has its own art and style. Some proofs are more elegant than others. For example, some proofs have all the given statements written at the beginning of the proof. Others will be guided by the postulates, placing given statements in the order they are needed. Another example of a good style feature is to have any statements that indicate addition or subtraction appear in consecutive lines. Example 2 Given: Prove: ΔFHG ≅ ΔGJI Solution: Strategy—Mark the two pairs of parallel lines with parallel markings. Then look for corresponding or alternate interior angles. ∠JIG and ∠HGF are congruent corresponding angles, as are ∠HFG and ∠JGI. After marking all these congruent pairs, we see that the AAS postulate applies.
Example 3 Given: ∠AWQ ≅ ∠WAQ, Q is the midpoint of Prove: ΔWPQ ≅ ΔASQ Solution: Strategy—Mark the figure with congruent markings, including because of the bisector. We have only a pair of congruent sides and right angles. However, ΔWQA has two congruent angles. So the triangle must be isosceles and . The HL postulate now applies.
Linear pairs or complementary angles that involve a congruent pair of angles sometimes lead us to a second congruent pair of angles. We can justify the second congruence statement with “angles supplementary/complementary to congruent angles are congruent.” Example 4 Given: Prove: ΔDAC ≅ ΔEBC Solution: Strategy—Mark the figure. We can see the two pairs of congruent sides from the givens. The middle triangle is isosceles, so we can mark ∠BAC ≅ ∠ABC. These angles are not part of the triangles we want to prove congruent. However, they form linear pairs with ∠DAC and ∠EBC, which lets us conclude that ∠DAC ≅ ∠EBC. The SAS postulate can be used.
Overlapping Triangles Certain techniques are helpful when the triangles you are trying to prove congruent overlap one another. Look for shared parts. Look for applications of the partition postulate, particularly along sides or angles where the triangles overlap. Sketch the figure with the triangles separated. Marking the separated figures can help you identify shared parts and the appropriate congruence postulate. If the two corresponding parts in the separated sketch have the same name, then they are shared and must be congruent. Angles, however, may be shared and congruent even if the points used to name the angle differ. The vertex must be the same. However, the second point for each ray may differ, as long as they lie on the same ray. In Figure 6.7, ∠ABE and ∠CBD are shared congruent angles.
Highlight each triangle in a different color. Doing this sometimes makes it easier to formulate a strategy. Overlapping triangles with a shared angle Example 1 Given: and Prove: ΔACD ≅ ΔPQR Solution: Strategy—Show using the addition property and the partition postulate. leads to . The parallel segments gives congruent corresponding angles. We can use the AAS postulate.
Example 2 Given: Prove: ΔRVT ≅ ΔRUS Solution: Strategy—Sketch the triangles separately to help identify the shared parts. ∠R is a shared angle. Also use segment addition to show . The marked congruent parts indicate SAS.
Overlapping angles can be handled in the same way as overlapping sides. With angles, however, a separated sketch is even more important as it can be difficult to visualize the two triangles when both angles and sides overlap. Of course, just because sides or angles overlap does not mean you will have to use segment or angle addition on those parts. In the following example, you need to use only angle addition. Math Fact Just because you sketch overlapping figures separately doesn’t mean you should not mark the original figure as well. Sometimes marking both is helpful. You may not see certain relationships like isosceles triangles, vertical angles, or linear pairs in the separated figure. Example 3 Given: ∠RTO ≅ ∠STA, Prove: ΔRTA ≅ ΔSTO Solution: Strategy—Sketch the separated triangles, and mark both the original and separated figures. The markings on the original show us that ΔRTS is isosceles; so ∠R ≅ ∠S. Then show ∠RTA ≅ ∠STO by angle addition. The separated sketch indicates that ASA will be used.
CPCTC Key Ideas Triangle congruence can be used to prove sides or angles are congruent. First identify a pair of triangles that contain the corresponding parts. Then prove the triangles are congruent. Then you can say that any pair of corresponding parts are congruent by CPCTC. Proofs do not always ask for triangles or other polygons to be proven congruent. You may be asked to prove a pair of sides or angles are congruent or to prove a particular relationship between sides or angles. You can often accomplish this by first proving a pair of polygons or triangles are congruent. Then applying CPCPC or CPCTC. CPCPC states that all corresponding parts of congruent polygons are congruent (CPCPC). The special case for triangles states that all corresponding parts of congruent triangles are congruent (CPCTC). When writing a proof involving CPCTC or CPCPC, you will need to identify the triangles or polygons to be proven congruent and then write an appropriate congruence statement before applying CPCTC or CPCPC. Be careful to list the vertices in the correct order when writing the congruence statement—the corresponding parts must match in the correct order. Example Given: and Prove: ∠O ≅ ∠P Solution: Strategy—first prove ΔMON ≅ MPN, then apply CPCTC
Proving Angle and Segment Relationships A proof may ask you to show that a segment or an angle has a certain property or is in a certain relationship with another segment or angle. CPCTC used with a congruence statement can help establish the desired relationship.
To prove a segment is a median—prove it divides the opposite side into congruent segments. To prove a segment is an angle bisector—prove it divides the angle into two congruent angles. To prove right angles—prove a linear pair of angles are congruent since two angles equal in measure that sum to 180° must measure 90° each. To prove a segment is an altitude—prove it forms right angles at the opposite side. Example 1 Given: Prove: is a median of ΔDRT Solution: Strategy—a median intersects the opposite side at its midpoint. We need to show . So the obvious approach would be to show ΔDAP ≅ ΔRAQ. The two pairs of parallel lines give two pairs of congruent corresponding angles, and the given sides let us use AAS. CPCTC then lets us conclude .
Example 2 Given: Prove: is an angle bisector of ∠XAR Solution: Strategy—prove ΔPAY ≅ ΔQAY, then use CPCTC to show that ∠PAY ≅ ∠QAY.
Double Congruence Proofs In some proofs, we may have to prove a first pair of triangles is congruent and then use CPCTC to establish the congruence of parts needed to prove a second pair of triangles is congruent. Example 1 Given: bisects ∠DCB Prove: ΔADE ≅ ΔABE Solution: Strategy—prove ΔADC ≅ ΔABC first, and then use CPCTC to prove and ∠DAE ≅ ∠EAB.
Example 2 Given: is a perpendicular bisector of ΔTRY, ∠S and ∠O are right angles, Prove: ΔYST ≅ ΔTOR Solution: Strategy—first prove ΔTMY ≅ ΔTMR, and then use CPCTC to show .
Proving Congruence by Transformations Key Ideas Figures can be proven congruent by identifying a sequence of rigid motions that map one onto the other. The angle and segment relationships between preimage, image, lines of reflection, center of rotation, and translation vectors can help establish specific transformations that map one point onto another. Congruent Triangles Two triangles are congruent if a sequence of rigid motions maps one triangle to another. Some clues to identifying rigid motions are: If the segments joining corresponding vertices of the two triangles are congruent and parallel, then one figure is a translation of the other. If a single line is the perpendicular bisector of segments formed by corresponding vertices, then one figure is the reflection of the other. If angles formed by corresponding vertices and a center point are all congruent, and corresponding distances to the center point are equal, then one figure is a rotation of the other. Example 1 Given: ΔWXY and perpendicular bisector Prove: ΔWZY ≅ ΔXZY using a sequence of rigid motions Solution: Strategy—the perpendicular bisector suggests that is a line of reflection. Show that one triangle is the image of the other after a reflection over . Proof—Point Y is the image of itself after a reflection over because Y lies on the line of reflection. Point Z is the image of itself for the same reason. X is the image of point W after a reflection over because a line of reflection is the perpendicular bisector of the segment joining the preimage and the image after the reflection. Therefore, and the triangles are congruent because one is mapped to the other by a reflection, which is a rigid motion. Example 2 Given: and Prove: ΔABC ≅ ΔDEF Solution: Point A translates to D by length AD. B translates to E and C translates to F by the same distances in a parallel direction. Therefore ΔABC is a translation of ΔDEF, and the triangles are congruent because translations are rigid motions.
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