{{ tocSubheader }}
{{ 'ml-label-loading-course' | message }}
An error ocurred, try again later!
Chapter {{ article.chapter.number }}
{{ article.number }}.
{{ article.displayTitle }}
{{ article.intro.summary }}
Show less Show more expand_more Lesson Settings & Tools
| | {{ 'ml-lesson-number-slides' | message : article.intro.bblockCount }} |
| | {{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount }} |
| | {{ 'ml-lesson-time-estimation' | message }} |
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 Angles
Consider the following statement.
Extra
How to Use the AppletIn the applet, rigid motions can be applied only on
- To translate select its interior region and slide.
- Point acts as the center of rotation and can be moved by dragging it.
- To rotate about click on any vertex of and drag it.
- The given line acts as a line of reflection which can be moved by dragging it. Similarly, its inclination can be changed by dragging either of its two points.
- To reflect across the line of reflection, push the
Reflect
button.
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 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
If two sides and the included angle of a triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
Based on the diagram above, the theorem can be written as follows.
Proof
Side-Angle-Side Congruence TheoremThis proof will be developed based on the given diagram, but it is valid for any pair of triangles.
1
Translate So That Two Corresponding Vertices Match
expand_more Translate so that is mapped onto If this translation maps onto the proof is complete.
Since the image of the translation does not match at least one more transformation is needed.
2
Rotate So That Two Corresponding Sides Match
expand_more Rotate counterclockwise about so that a pair of corresponding sides match. If the image of this transformation is the proof is complete. Note that this rotation maps onto Therefore, the rotation maps onto
As before, the image does not match Therefore, a third rigid motion is required.
3
Reflect So That Two More Corresponding Sides Match
expand_more Reflect across Because reflections preserve angles, is mapped onto Additionally, it is given that Therefore, is mapped onto which gives that is mapped onto
This time the image matches
Example
Identifying Congruent Triangles
In the following diagram, triangles and are congruent, and is congruent to
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.
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, and
Since these triangles are congruent, their corresponding parts are congruent. This implies that is congruent to
First Pair
Because and are parts of and respectively, consider triangles and
Second Pair
Next, consider triangles and Because and are congruent, and are congruent. Additionally, since and are congruent, is congruent to
Third Pair
The last two triangles to consider are triangles and 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 and the rays and to construct two different triangles, one at a time, in such a way that the following conditions are met.
- The angle formed at has the same measure in both triangles.
- The angle formed at 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.
If two angles and the included side of a triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
Based on the diagram above, the theorem can be written as follows.
Proof
Angle-Side-Angle Congruent TheoremThis proof will be developed based on the given diagram, but it is valid for any pair of triangles.
1
Translate So That Two Corresponding Vertices Match
expand_more Translate so that is mapped onto If this translation maps onto the proof is complete.
Since the image of the translation does not match at least one more transformation is needed.
2
Rotate So That Two Corresponding Sides Match
expand_more Rotate counterclockwise about so that a pair of corresponding sides match. If the image of this transformation is the proof is complete. Note that this rotation maps onto Therefore, the rotation maps onto
As before, the image does not match Therefore, a third rigid motion is required.
3
Reflect So That All Corresponding Sides Match
expand_more Reflect across Because reflections preserve angles, and are mapped onto and respectively. Then, the point of intersection of the original rays is mapped onto the point of intersection of the image rays
This time the image matches
Example
Write Equations Based on Congruent Triangles
Consider the following diagram.
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["9.75"]}}
Hint
Take note that 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 is a common side for triangles and
By the Reflexive Property of Congruence, is congruent to itself. Additionally, and are congruent, as are and
Consequently, and are congruent because of the Angle-Side-Angle (ASA) Congruence Theorem. Therefore, the corresponding sides and angles are congruent.
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.
By solving Equation (I), the value of can be found.
Next, solve Equation (III) to find the value of
Then, the value of can be found by substituting into the Equation (II) and solving the resulting equation for
Finally, the required sum can be calculated by substituting the values found for and
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 and 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.
Discussion
Side-Side-Side Congruence Theorem
As seen in the previous exploration, the Side-Side-Side is a valid criterion for checking triangle congruence.
If the three sides of a triangle are congruent to the three sides of another triangle, then the triangles are congruent.
Based on the diagram above, the theorem can be written as follows.
Proof
Side-Side-Side Congruence TheoremThis proof will be developed based on the given diagram, but it is valid for any pair of triangles.
1
Translate So That Two Corresponding Vertices Match
expand_more Translate so that is mapped onto If this translation maps onto the proof is complete.
Since the image of the translation does not match at least one more transformation is needed.
2
Rotate So That Two Corresponding Sides Match
expand_more Rotate counterclockwise about so that a pair of corresponding sides matches. If the image of this transformation is the proof is complete. Note that this rotation maps onto Consequently, is mapped onto
As before, the image does not match Therefore, a third rigid motion is required.
3
Reflect So That Two More Corresponding Sides Match
expand_more The points and are on opposite sides of Now, consider Let denote the point of intersection between and
It can be noted that and By the Converse Perpendicular Bisector Theorem, is a perpendicular bisector of Points along the perpendicular bisector are equidistant from the endpoints of the segment, so
Given three random segments, it is not always possible to construct a triangle. But, when possible, this triangle will be unique. This fact implies that the angle measures of that triangle are also unique.
Example
Using Triangle Congruence Conditions to Solve Problems
In the following diagram, is a rectangle, and are squares, and are isosceles triangles, is congruent to and is congruent to
{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"rendered":false},"text":""},"formTextBefore":null,"formTextAfter":null,"answer":{"text":["70"]}}
Hint
Using the Segment Addition Postulate and the Side-Side-Side (SSS) Congruence Theorem, prove that is congruent to Then, find the measure of Use the fact that
Solution
To find the value of notice that and add up
Since is a rectangle and is a square, and are right angles. Therefore, these angles have a measure of each. Also, it is given that Use this information to solve for
An expression of was found in terms of Therefore, to find the value of the value of should first be known. 
Since is congruent to these segments have the same length, that is, Simlarly, By substituting these expressions into Equation (I), a relation between and will be obtained.
This equation implies that and are congruent.
Because is isosceles, is congruent to Therefore, Keep in mind that it is given that and are congruent. That means These two expressions can be substituted into Equation (I).
Based on the equation just obtained, it can be concluded that and are congruent.
The Side-Side-Side (SSS) Congruence Theorem allows to conclude that is congruent to
Since corresponding parts of congruent triangles are congruent, it can be concluded that is congruent to Therefore,
Finding the Value of
If and can be proven to be congruent, that would provide the needed information to find the value of Therefore, focus on those two triangles.
Proving that
Notice that is common to both triangles. By using the Segment Addition Postulate, the following pair of equations can be written.
Proving that
Once more, the Segment Addition Postulate can be used to rewrite and
Proving that
Since is a rectangle, and are squares, and and are isosceles triangles, the following consequences can be drawn.
| Given | Consequence |
|---|---|
| is a rectangle | |
| is a square | |
| is a square | |
| is an isosceles triangle | |
| is an isosceles triangle |
Next, organize the information in the right-hand column in a flow chart and use the Transitive Property of Congruence to prove that
Proving that
Previously, the following three congruence statements were obtained.Finding the Value of
Finally, to find the value of substitute into the equation Then solve forExplore
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 and the rays and to construct two different triangles, one at a time, in such a way that these conditions are met:
- The angle formed at has the same measure in both triangles.
- The angle formed at 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.
If two angles and a non-included side of a triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.
Based on the diagram above, the theorem can be written as follows.
Proof
This proof will be developed based on the given diagram, but it is valid for any pair of triangles.
1
Translate So That Two Corresponding Vertices Match
expand_more Translate so that is mapped onto If this translation maps onto the proof is complete.
Since the image of the translation does not match at least one more transformation is needed.
2
Rotate So That Two Corresponding Sides Match
expand_more Rotate clockwise about so that a pair of corresponding sides match. If the image of this transformation is the proof is complete. Note that this rotation maps onto Therefore, the rotation maps onto
As before, the image does not match Therefore, a third rigid motion is required.
3
Reflect So That Two More Corresponding Sides Match
expand_more It is given that two angles of are congruent to two angles of Hence, by the Third Angle Theorem, is congruent to
Example
Proving Congruence in Triangles
Dylan bought a new boomerang to play with his friends next summer. In the drawing printed on the boomerang, and are congruent, and and are congruent.
Show that is congruent to
Answer
See solution.
Hint
Separate triangles and and notice they have a common angle. Then, use the Angle-Angle-Side (AAS) Congruence Theorem.
Solution
Start by separating and from the design.
Notice that is common to both triangles. By the Reflexive Property of Congruence, is congruent to itself. Also, it is given that is congruent to and is congruent to
Applying the Angle-Angle-Side (AAS) Congruence Theorem, it is obtained that is congruent to Consequently, their corresponding parts are congruent, which means that is congruent to
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 and to construct two different triangles in such a way that the angle formed at has the same measure in both triangles.
Closure
Side-Side-Angle Congruence Theorem for Right Triangles
With the previous applet, it can be checked that, in general, the Side-Side-Angle is not a valid criterion to determine triangle congruence. For instance, the following triangles meet the conditions of this criterion, and they are not congruent.
However, this criteria is valid in the particular case that both triangles are right triangles.
Rule
Hypotenuse Leg Theorem
If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent.
Based on the diagram, the following relations hold true.
Proof
Consider and shown below.
Therefore, by the Side-Side-Side Congruence Theorem the triangles are congruent.
In the following chart, all the criteria for triangle congruence seen in the lesson are listed.
Edit Lesson
Loading content