8. Congruent Triangles
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Chapter 1
8.
Congruent Triangles
Congruent triangles are a fundamental concept. They refer to triangles that are identical in shape and size. Although they might appear different based on their position or orientation, every corresponding side and angle of these triangles are equal. Recognizing and proving the congruency of triangles is vital, as it plays a pivotal role in various mathematical proofs and real-world scenarios. Whether in architectural designs, art, or nature, congruent triangles offer symmetry and balance, making them an essential geometrical concept to grasp.
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| | 9 Theory slides |
| | 9 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Congruent Triangles
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According to the definition of congruence, two figures are congruent if there is a rigid motion that maps one onto the other. In comparison, for polygons, there is another method that makes it possible to determine congruence without involving rigid motions. That method will be developed and understood in this lesson.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Determining If Two Triangles Are Congruent
By applying rigid motions, investigate whether the given pair of triangles are congruent or not.
- The left-hand side triangle can be translated by dragging it.
- The left-hand side triangle can be rotated about the movable point P by clicking and dragging any of its vertices.
- The left-hand side triangle can be reflected across the line shown, which can be set by moving its points.
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Discussion
Accounting For Corresponding Parts
When a rigid motion is applied to a polygon, the resulting image can be seen as the polygon formed by the image of each part of the original polygon under the same rigid motion.
Concept
Corresponding Parts
Consider two figures. One figure is the image of the other under a transformation. The pairs formed by a part of the preimage — a side, angle, or vertex — and the image of that part are called corresponding parts. For example, in the following applet, A and P are corresponding vertices.
| Congruent or Similar? | Relationship |
|---|---|
| Congruent | The corresponding sides and angles of the figures are congruent. |
| Similar | The corresponding sides of the figures are proportional.
The corresponding angles of the figures are also congruent. |
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Discussion
Properties of Congruent Triangles
When two triangles are congruent, it seems natural to think that the corresponding parts are congruent as well. What about the same concept when the statement is reversed? That is to say, if the corresponding parts of two triangles are congruent, are the triangles congruent? The following claim answers this question.
Rule
Congruent Triangles
Two triangles are congruent if and only if their corresponding sides and angles are congruent.
Using the triangles shown, this claim can be written algebraically as follows.
△ ABC ≅ △ DEF ⇕ AB≅DE BC≅EF AC≅DF and ∠A≅∠D ∠B≅∠E ∠C≅∠F
Proof
This proof will be developed based on the given diagram, but it is valid for any pair of triangles. The proof of this biconditional statement consists of two parts, one for each direction.
- If △ ABC and △ DEF are congruent, then their corresponding sides and angles are congruent.
- If the corresponding sides and angles of △ ABC and △ DEF are congruent, then the triangles are congruent.
Part 1
By definition of congruent figures, if the triangles are congruent there is a rigid motion or sequence of rigid motions that maps △ ABC onto △ DEF.
Because rigid motions preserve side lengths, AB and its image have the same length, that is, AB=DE. Therefore, AB≅DE. Similarly for the other two side lengths. BC≅EF and AC≅DF Furthermore, rigid motions preserve angle measures. Then, ∠ A and its image have the same measure, that is, m∠ A = m∠ D. Therefore, ∠ A ≅ ∠ D. Similarly for the remaining angles. ∠ B ≅ ∠ E and ∠ C ≅ ∠ F That way, it has been shown that if two triangles are congruent, then their corresponding sides and angles are congruent.
Part 2
To begin, mark the congruent parts on the given diagram.
The primary purpose is finding a rigid motion or sequence of rigid motions that maps one triangle onto the other. This can be done in several ways, here it is shown one of them.
1
Translate △ ABC so that one pair of corresponding vertices match
expand_more Apply a translation to △ ABC that maps A to D. If this translation maps △ ABC onto △ DEF the proof will be complete.
As seen, △ A'B'C' did not match △ DEF. Therefore, a second rigid motion is needed.
2
Rotate △ DB'C' so that one pair of corresponding sides match
expand_more Apply a clockwise rotation to △ DB'C' about D through ∠ EDB'. If the image matches △ DEF, the proof will be complete. Notice this rotation maps B' onto E, and therefore, DB' onto DE.
As before, the image did not match △ DEF. Thus, a third rigid motion is required.
3
Reflect △ DEC'' so that the corresponding sides match
expand_more Apply a reflection to △ DEC'' across DE. Because reflections preserve angles, DC'' is mapped onto DF and EC'' is mapped onto EF. Then, the intersection of the original rays C'', is mapped to the intersection of the image rays F.
This time the image matched △ DEF.
Consequently, through applying different rigid motions, △ ABC was mapped onto △ DEF. This implies that △ ABC and △ DEF are congruent. Then, the proof is complete.
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Example
Investigating If Two Triangles Are Congruent
The local chess team in Jordan's town has designed a fancy new logo.
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What is the sum of the perimeters of both triangles?
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Hint
Use the measuring tool to find the side lengths and angle measures of each triangle. Then, compare them. Are all corresponding parts congruent? If the triangles are congruent, they have equal perimeters.
Solution
To determine whether the logo is made of two congruent triangles, the measures of both triangles need to be found. For simplicity, start by separating the two triangles involved.
Then, find and label the side lengths and angle measures of each triangle.
As seen, both triangles have the same side lengths and angle measures. That is, all their corresponding sides and angles are congruent. Therefore, the two triangles are congruent. Jordan was correct!
The new logo is made of two congruent triangles.
Because the triangles are congruent, they have the same perimeter. Then, it is enough to find the perimeter of only one of them and then multiply it by 2. Remember, the perimeter is the sum of the side lengths. P_1 &= 1.5+3+3.4 P_1 &= 7.9 Finally, the sum of the perimeters of both triangles is twice P_1. That can be expressed as 2* P_1 = 2*7.9 = 15.8.
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Example
Using Properties of Triangles to Determine Congruence
To show that two triangles are not congruent, it is enough to find only one side or angle of one of the triangles that is not congruent to any side or angle of the second triangle.
Mark, Magdalena, and Kevin each bought a notebook with a triangle design on the cover.
Are there any pairs of congruent triangles? If so, which ones are congruent?
Answer
No, there are no pairs of congruent triangles.
Hint
To conclude that two triangles are congruent, check that all corresponding sides and angles are congruent.
Solution
Beginning with Mark's notebook, notice that the triangle has a 50^(∘) angle, while neither of the other two triangles have a 50^(∘) angle. That is, there is a part in Mark's triangle that is not congruent to any part of the other two triangles.
The triangle in Mark's notebook is not congruent to any of the other triangles.
Now, moving on to Madaelena's notebeook. Notice that the triangle has the same angle measures as the triangle in Kevin's notebook.
Before concluding that these two triangles are congruent, remember, all the corresponding parts have to be congruent. Therefore, the sides opposite to the 80^(∘) angle of each triangle should be congruent. Well, that is not the case.
It was a close call, but the triangle in Magdalena's notebook is not congruent to the triangle in Kevin's notebook. As a result, none of the triangles are congruent to the others.
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Pop Quiz
Practice Identifying Congruent Triangles
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Example
Investigating Congruent Triangles
The following figure is made of 5 triangles.
How many pairs of congruent triangles are there?
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Answer
There are four pairs of congruent triangles in the given figure.
△ ABF &≅ △ DFB △ AFG &≅ △ DEF △ AFG &≅ △ BDC △ DEF &≅ △ BDC
Hint
Notice that BF is a common side for △ ABF and △ DFB. Remember, if two sides or angles have the same number of marks, they are congruent.
Solution
To determine which pairs of triangles are congruent, pick one of them and compare its parts to the parts of the rest of the triangles. Start by comparing △ AFG to △ ABF. These two triangles have AF as a common side.
By applying the concept of reading the marks to check for congruency, it can be said that BF is not congruent to any side of △ AFG. Consequently, △ AFG is not congruent to △ ABF. By the same reasoning, △ AFG is not congruent to △ BFD either. △ AFG ≆ △ ABF △ AFG ≆ △ BFD Next, compare △ AFG to △ DEF.
Since ∠ G and ∠ DFE have both a measure of 50^(∘), they are congruent angles. Furthermore, the rest of the corresponding parts of these two triangles are congruent. ccc AF ≅ DE & & ∠ GAF ≅ ∠ FDE FG ≅ EF & and & ∠ AFG ≅ ∠ DEF AG ≅ DF & & ∠ FGA ≅ ∠ EFD Consequently, △ AFG and △ DEF are congruent. Similarly, it can be checked that △ AFG is also congruent to △ BDC. By the Transitive Property of Congruence, a third pair of congruent triangles can be obtained.
Next, compare △ ABF to △ DFB. Since ∠ AFB and ∠ DBF have the same measure, they are congruent. Additionally, BF is a common side for both triangles and, by the Reflexive Property of Congruence, BF≅ BF. BF≅ BF and ∠ AFB ≅ ∠ DBF In addition, the rest of corresponding parts of these two triangles are congruent. ccc AB ≅ DF && ∠ FAB ≅ ∠ BDF BF ≅ FB &and& ∠ ABF ≅ ∠ DFB AF ≅ DB && ∠ BFA ≅ ∠ FBD Consequently, △ ABF and △ DFB are congruent. So far, there are four pairs of congruent triangles. △ ABF &≅ △ DFB △ AFG &≅ △ DEF △ AFG &≅ △ BDC △ DEF &≅ △ BDC Taking into account the Transitive Property of Congruence, and the fact that △ ABF is not congruent to △ AFG, it can be said that there are no more pairs of congruent triangles.
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Closure
Writing a Congruence Statement
As with polygons, when it comes to writing a triangle congruence statement, the order in which the vertices are written is critical. Naming them in an incorrect order leads to erroneous conclusions. Consider, for example, the following congruent triangles.
Although the triangles are congruent, the congruence statement △ ABC ≅ △ MQT is incorrect. Why? Because it would lead to the following conclusions about the sides and angles.
|
Two polygons are congruent if and only if their corresponding sides and angles are congruent. |
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Congruent Triangles
Exercise 1.1
Consider the following pair of triangles.
Dylan has written four different congruence statements on a piece of paper.
One of them is wrong. Which one?
{"type":"choice","form":{"alts":["#1","#2","#3","#4"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":2}
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When describing congruent shapes, it is important when we write a congruence statement that the corresponding vertices come in the same order. In congruent triangles, vertices that are between the same two pairs of corresponding sides are corresponding. With this information, we can identify corresponding vertices.
Now we can write a congruence statement. △ABC ≅ △ FDE [-1em] A→ F, B → D, C→ E Immediately we can see that the first statement is correct. Let's color the vertices in the next three statement to check in which ones the letters are paired correctly. #2 △ BAC ≅ DFE ✓ #3 △ E DF ≅ C AB * #4 △ CAB ≅ EFD ✓ In the third statement, the letters are not paired correctly.
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Solution
{"type":"pair","form":{"alts":[[{"id":0,"text":"∠ A"},{"id":1,"text":"∠ B"},{"id":2,"text":"∠ C"}],[{"id":0,"text":"∠ E"},{"id":1,"text":"∠ F"},{"id":2,"text":"∠ D"}]],"lockLeft":true,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[0,1,2],[0,1,2]]}
{"type":"pair","form":{"alts":[[{"id":0,"text":"AB"},{"id":1,"text":"BC"},{"id":2,"text":"CA"}],[{"id":0,"text":"EF"},{"id":1,"text":"FD"},{"id":2,"text":"DE"}]],"lockLeft":true,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[0,1,2],[0,1,2]]}
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In congruent triangles, corresponding vertices have the same angle measures. Examining the diagram, we can identify which angles that are congruent by counting the number of angle markers. For example both ∠ B and ∠ F have three angle markers and are therefore congruent.
Let's identify the remaining corresponding vertices as well.
Let's pair the angles. ∠ A ≅ ∠ E ∠ B ≅ ∠ F ∠ C ≅ ∠ D
Corresponding sides are the sides in two congruent shapes that are between the same two pairs of corresponding vertices. For example in △ ABC, we see that BC is between the angles with two and three markers respectively. In △ EFD we see that DF is also between the angles with two and three markers respectively. Therefore, BC and DF are corresponding sides.
Let's identify the remaining corresponding sides in the same way.
Let's pair the sides. AB ≅ EF BC ≅ FD CA ≅ DE
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Solution
In the following diagram, △ XYZ≅ △ ONL.
Complete each of the following statements. m∠ Y= □ [1em] m∠ O= □ [1em] m∠ Z= □
{"type":"pair","form":{"alts":[[{"id":0,"text":"m∠ Y"},{"id":1,"text":"m∠ O"},{"id":2,"text":"m∠ Z"}],[{"id":0,"text":"135^(∘)"},{"id":1,"text":"30^(∘)"},{"id":2,"text":"15^(∘)"}]],"lockLeft":true,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[0,1,2],[0,1,2]]}
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Let's start by identifying the corresponding vertices by analyzing the congruence statement. △XYZ ≅ △ ONL [-1em] X→ O, Y → N, Z→ L Let's mark corresponding vertices and consequently which angles are congruent.
Now we know both m∠ Y and m∠ O. We can determine m∠ Z by using the Interior Angles Theorem. m∠ Z+30^(∘)+135^(∘) = 180^(∘) ⇓ m∠ Z=15^(∘) Now we can complete the statements. m∠ Y&=$135^(∘)$ [0.5em] m∠ O&=$30^(∘)$ [0.5em] m∠ Z&=$15^(∘)$
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Solution
Find the measure of ∠ R if △ ABC≅ △ SQR.
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Let's use the congruence statement to identify corresponding angles. △ABC ≅ △ SQR [-1em] A→ S, B → Q, C→ R With this information, we can identify the measures of ∠ S and ∠ Q as being 45^(∘) and 80^(∘) respectively.
Now we can find m∠ R by using the Interior Angles Theorem. 45^(∘)+80^(∘)+m∠ R = 180^(∘) ⇓ m∠ R = 55^(∘)
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Solution
Consider the following triangles.
Complete each of the following statements. XY= [1em] XZ= [1em] YZ=
{"type":"pair","form":{"alts":[[{"id":0,"text":"XY="},{"id":1,"text":"XZ="},{"id":2,"text":"YZ="}],[{"id":0,"text":"4"},{"id":1,"text":"3.8"},{"id":2,"text":"5.5"}]],"lockLeft":true,"lockRight":false},"formTextBefore":"","formTextAfter":"","answer":[[0,1,2],[0,1,2]]}
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Using the congruence statement, we can match corresponding vertices.
Now we can identify the lengths of the different sides in △ XZY.
We can now complete the statements. XY&=$4$ [0.5em] XZ&=$3.8$ [0.5em] YZ&=$5.5$
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Solution
Classmates Ali and Magdalena are investigating the relationship between two triangles. Ali claims that △ ABC≅ △ YXZ while Magdalena says that △ ZXY≅ △ CBA. Who is correct?
{"type":"choice","form":{"alts":["Ali is correct","Magdalena is correct","Both are correct","Neither is correct"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":2}
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Let's identify corresponding vertices and mark them with the same color.
A correct congruence statement states corresponding vertices in the same order. With this in mind, let's check both statements.
| Person | Congruence Statement | Correct order? | Valid statement? |
|---|---|---|---|
| Ali | △ A B C ≅△ Y XZ | Yes | Yes |
| Magdalena | △ Z XY ≅ △ C BA | Yes | Yes |
Both are correct.
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Solution
Congruent Triangles
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