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4.4 C) Congruent Triangles – Part 2
4.4 C) Congruent Triangles – Part 2
In the previous section, we learnt that there are 4 different ways that we can prove that triangles are congruent to one another:
The previous section has more information on the ways of proving congruent triangles (click here to be taken back to the previous section).
- Side-Side-Side (SSS)
- Angle-Angle-Side (AAS)
- Side-Angle-Side (SAS)
- Right-angle-Hypotenuse-Side (RHS)
The previous section has more information on the ways of proving congruent triangles (click here to be taken back to the previous section).
Examples 1
The triangle ABC is an equilateral triangle. The line AD is perpendicular to the line BC and it meets the line BC at point D.
The triangle ABC is an equilateral triangle. The line AD is perpendicular to the line BC and it meets the line BC at point D.
Prove that triangle ABD is congruent to triangle ADC.
From the above sketch, we can see that we have two right angles; angle ADC and angle ADB are both right angles. These angles are right angles because the line AD is perpendicular to the line BC. Perpendicular means that the lines meet at 90°, thus meaning that angle ADC and angle ADB are 90°. As we have two right-angle triangles, we are probably going to be using the RHS rule. The R in this rule stands for right-angle, which we have in both triangles. The H stands for hypotenuse and the hypotenuse is the longest side of the triangle. The hypotenuse for triangle ABD is AB and the hypotenuse for triangle ADC is AC. The lines AB and AC form the outside of the equilateral triangles. Equilateral triangles have equal sides, and this means that AB will be equal to AC; we have equal hypotenuses.
From the above sketch, we can see that we have two right angles; angle ADC and angle ADB are both right angles. These angles are right angles because the line AD is perpendicular to the line BC. Perpendicular means that the lines meet at 90°, thus meaning that angle ADC and angle ADB are 90°. As we have two right-angle triangles, we are probably going to be using the RHS rule. The R in this rule stands for right-angle, which we have in both triangles. The H stands for hypotenuse and the hypotenuse is the longest side of the triangle. The hypotenuse for triangle ABD is AB and the hypotenuse for triangle ADC is AC. The lines AB and AC form the outside of the equilateral triangles. Equilateral triangles have equal sides, and this means that AB will be equal to AC; we have equal hypotenuses.
The S in RHS stands for corresponding sides. Both of the triangles, ABD and ADC, have a common side of AD.
Therefore, because of RHS, triangles ABD and ADC are congruent.
My working has been very wordy. You would not be expected to write this much in the exam. You would only need to write what is shown below.
Therefore, because of RHS, triangles ABD and ADC are congruent.
My working has been very wordy. You would not be expected to write this much in the exam. You would only need to write what is shown below.
Example 2
The lengths AB, BC, CD and DA on the shape below are all equal.
The lengths AB, BC, CD and DA on the shape below are all equal.
Prove that triangle ABC and ACD are congruent.
From the question, we are told that the lengths AB, BC, CD and DA are all the same length. This means that the length AB is equal to AD, and the length BC is equal to DC. The other side in both of the triangles is AB; we have a common side. This means that we have three pairs of equal sides and due to the SSS rule, these triangles are congruent to each other.
From the question, we are told that the lengths AB, BC, CD and DA are all the same length. This means that the length AB is equal to AD, and the length BC is equal to DC. The other side in both of the triangles is AB; we have a common side. This means that we have three pairs of equal sides and due to the SSS rule, these triangles are congruent to each other.
Example 3
The shape ABCD is a parallelogram.
The shape ABCD is a parallelogram.
Prove that the triangle ABC and ACD are congruent.
Before we start proving that these two triangles are congruent to one another, I am going to quickly go over the rules for parallelograms. Opposites sides in parallelograms are parallel to one another and equal in length. This means that for the above parallelogram, the sides BC and AD are equal in length, and the sides AB and DC are equal in length.
We are trying to prove that ABC is congruent to ACD. We have already established that BC is equal to AD, and AB is equal to DC. This means that we have two sides in each of the triangles that are the same. The final side in both of the triangles is AC (AC is common).
Therefore, all 3 of the sides in the triangles are the same length and this means that the triangles are congruent to one another because of the SSS rule.
Before we start proving that these two triangles are congruent to one another, I am going to quickly go over the rules for parallelograms. Opposites sides in parallelograms are parallel to one another and equal in length. This means that for the above parallelogram, the sides BC and AD are equal in length, and the sides AB and DC are equal in length.
We are trying to prove that ABC is congruent to ACD. We have already established that BC is equal to AD, and AB is equal to DC. This means that we have two sides in each of the triangles that are the same. The final side in both of the triangles is AC (AC is common).
Therefore, all 3 of the sides in the triangles are the same length and this means that the triangles are congruent to one another because of the SSS rule.
Example 4
The lines AB and ED are parallel to each other and equal in length. The lines AD and BE intercept one another at point C.
The lines AB and ED are parallel to each other and equal in length. The lines AD and BE intercept one another at point C.
Prove that triangle ABC and CDE are congruent.
We are told in the question that the line AB and ED are equal to one another; we have one side in the triangles that is equal.
We are also told that the lines AB and ED are parallel to one another. We are able to use the parallel line rules to work out the angles in the triangle. One of the parallel line rules is alternate angles (z angles). The z that I am going to create is from ABED and it is drawn in green on the diagram below.
We are told in the question that the line AB and ED are equal to one another; we have one side in the triangles that is equal.
We are also told that the lines AB and ED are parallel to one another. We are able to use the parallel line rules to work out the angles in the triangle. One of the parallel line rules is alternate angles (z angles). The z that I am going to create is from ABED and it is drawn in green on the diagram below.
The alternate angle rule means that angle ABC and angle CED are equal to one another.
There is another set of alternate angles in the shape. The second set of alternate angles comes from the z BADE. This z is shown in orange on the diagram below.
There is another set of alternate angles in the shape. The second set of alternate angles comes from the z BADE. This z is shown in orange on the diagram below.
This means that angle BAC and CDE are equal to one another.
We now have two angles and a corresponding side that are equal in each of the triangles. Therefore, we can say that these triangles are congruent due to the Angle-Angle-Side rule (AAS).
We now have two angles and a corresponding side that are equal in each of the triangles. Therefore, we can say that these triangles are congruent due to the Angle-Angle-Side rule (AAS).