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Two triangles are congruent if their corresponding sides and angles are congruent. However, there could be cases where not all side lengths or angle measures are known. The good news is that congruence can still be verified depending on which parts are known.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Explore
Investigating Congruence of Triangles Given Two Sides and the Included Angle
In the previous exploration, it was seen that a pair of triangles can have corresponding congruent angles but not be congruent triangles. Therefore, relying only on the relationship of only angles is not a valid criterion.
Angle-Angle-Angle is not a valid criterion for proving triangle congruence.
Discussion
Side-Angle-Side Congruence Theorem
The previous exploration suggests that two triangles are congruent whenever they have two pairs of corresponding congruent sides and the corresponding included angles are congruent. In fact, this conclusion is formalized in the Side-Angle-Side Congruence Theorem
Example
Identifying Congruent Triangles
In the following diagram, triangles ADE and BCE are congruent, and ∠ADC is congruent to ∠BCD.
How many more pairs of congruent triangles are there in the diagram? Name each congruent triangle pair.
Answer
There are two more pairs of congruent triangles.
△ADC△ABD≅△BCD≅△BAC
Hint
Remember, if two triangles are congruent, then their corresponding sides and angles are congruent.
Solution
Start by highlighting the given pair of congruent triangles, △ADE and △BCE.
Since these triangles are congruent, their corresponding parts are congruent. This implies that AD is congruent to BC.
First Pair
Because △ADE and △BCE are parts of △ADC and △BCD respectively, consider triangles ADC and BCD.
AD≅BC∠ADC≅∠BCDDC≅CDSideAngleSide
By the Side-Angle-Side (SAS) Congruence Theorem, it can be concluded that △ADC and △BCD are congruent. △ADC≅△BCD
Second Pair
Next, consider triangles ABD and BAC. Because △ADE and △BCE are congruent, ∠ADB and ∠BCA are congruent. Additionally, since △ADC and △BCD are congruent, CA is congruent to DB.
AD≅BC∠ADB≅∠BCADB≅CASideAngleSide
One more time, the Side-Angle-Side (SAS) Congruence Theorem can be used to conclude that triangles ABD and BAC are congruent. △ABD≅△BAC
Third Pair
The last two triangles to consider are triangles ABE and DEC. Unlike the first two pairs, these dimensions seem to be quite different. Therefore, it can be concluded that they are not congruent.
Consequently, in the initial diagram, there are two more pairs of congruent triangles in addition to the given one.
Explore
Investigating Congruence of Triangles Given Two Angles and the Included Side
Use segment AB and the rays AX and BY to construct two different triangles, one at a time, in such a way that the following conditions are met.
- The angle formed at A has the same measure in both triangles.
- The angle formed at B has the same measure in both triangles.
Discussion
Angle-Side-Angle Congruence Theorem
The following statement could be seen in the previous applet. When two triangles have two pairs of corresponding congruent angles, and the included corresponding sides are congruent, the triangles are then congruent. That leads to the second criteria for triangle congruence.
Example
Write Equations Based on Congruent Triangles
Consider the following diagram.
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Hint
Take note that QS is a common side for two triangles. Use the fact that if two triangles are congruent, their corresponding sides and angles are congruent.
Solution
Notice that QS is a common side for triangles PQS and RSQ.
By the Reflexive Property of Congruence, QS is congruent to itself. Additionally, ∠PQS and ∠RSQ are congruent, as are ∠PSQ and ∠RQS.
Finally, the required sum can be calculated by substituting the values found for x, y, and z.
∠PQS≅∠RSQQS≅QS∠PSQ≅∠RQSAngleSideAngle
Consequently, △PQS and △RSQ are congruent because of the Angle-Side-Angle (ASA) Congruence Theorem. Therefore, the corresponding sides and angles are congruent.
△PQS≅△RSQ⇒⎩⎪⎪⎨⎪⎪⎧∠PPQPS≅∠R≅RS≅RQ
By definition, congruent angles have the same measure, and congruent segments have the same length. Therefore, the congruence statements on the right-hand side support the formation of the following three equations.
⎩⎪⎪⎨⎪⎪⎧8z+2=506=2y+x5x−3=4.5(I)(II)(III)
By solving Equation (I), the value of z can be found.
Next, solve Equation (III) to find the value of x.
Then, the value of y can be found by substituting x=1.5 into the Equation (II) and solving the resulting equation for y.
6=2y+x
Substitute
x=1.5
6=2y+1.5
Solve for y
SubEqn
LHS−1.5=RHS−1.5
4.5=2y
DivEqn
LHS/2=RHS/2
24.5=y
UseCalc
Use a calculator
2.25=y
RearrangeEqn
Rearrange equation
y=2.25
Explore
Congruence of Triangles Given Three Pairs of Congruent Sides
At the beginning of the lesson, it was shown that the Angle-Angle-Angle is not a valid criterion for determining triangle congruence. Next, using the following applet, it will be investigated if the Side-Side-Side is a valid criterion. Use segments AB, AC, and BC to construct two different triangles. Construct the triangles one at a time.
Once the two triangles are drawn, find the angle measures of each triangle. Is there any relationship between the triangles? Repeat the process a couple of times to see if the relationship holds true.
Explore
Congruence of Triangles Given Two Angles and a Nonincluded Side
Notice that the ASA criterion requires the congruent sides to be included between the two pairs of corresponding congruent angles. Using the following applet, investigate what happens when the congruent sides are not the included sides.
Use segment AB and the rays AX and BY to construct two different triangles, one at a time, in such a way that these conditions are met:
- The angle formed at A has the same measure in both triangles.
- The angle formed at C, the intersection of the rays, has the same measure in both triangles.
Discussion
Angle-Angle-Side Congruence Theorem
As seen in the previous exploration, the Angle-Angle-Side condition is a valid criterion for triangle congruence.
Example
Proving Congruence in Triangles
Dylan bought a new boomerang to play with his friends next summer. In the drawing printed on the boomerang, ∠A and ∠C are congruent, and BF and BE are congruent.
Show that AE is congruent to CF.
Answer
See solution.
Hint
Separate triangles ABE and CBF and notice they have a common angle. Then, use the Angle-Angle-Side (AAS) Congruence Theorem.
Solution
Start by separating △ABE and △CBF from the design.
Notice that ∠B is common to both triangles. By the Reflexive Property of Congruence, ∠B is congruent to itself. Also, it is given that ∠A is congruent to ∠C, and BF is congruent to BE.
∠A≅∠C∠B≅∠BBF≅BEAngleAngleSide
Applying the Angle-Angle-Side (AAS) Congruence Theorem, it is obtained that △ABE is congruent to △CBF. Consequently, their corresponding parts are congruent, which means that AE is congruent to CF.
Explore
Investigating Side-Side-Angle
With the help of the following applet, investigate if the Side-Side-Angle is a valid criterion for determining triangle congruence.
Use segments AB and AC to construct two different triangles in such a way that the angle formed at B has the same measure in both triangles.