Max constructs a triangle in Geogebra. He tells Alicia that his triangle has a \begin{align*}42^\circ\end{align*} angle, a side of length 12, and a side of length 8. With only this information, will Alicia be able to construct a triangle that must be congruent to Max's triangle?
Watch This
http://www.youtube.com/watch?v=CA1TvVRApkQ James Sousa: Introduction to Congruent Triangles
Guidance
Two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs are congruent. While one way to show that two triangles are congruent is to verify that all side and angle pairs are congruent, there are five “shortcuts”. The following list summarizes the different criteria that can be used to show triangle congruence.
- AAS (Angle-Angle-Side): If two triangles have two pairs of congruent angles and a non-common side of the angles in one triangle is congruent to the corresponding side in the other triangle, then the triangles are congruent.
- ASA (Angle-Side-Angle): If two triangles have two pairs of congruent angles and the common side of the angles in one triangle is congruent to the corresponding side in the other triangle, then the triangles are congruent.
- SAS (Side-Angle-Side): If two triangles have two pairs of congruent sides and the included angle in one triangle is congruent to the included angle in the other triangle, then the triangles are congruent.
- SSS (Side-Side-Side): If two triangles have three pairs of congruent sides, then the triangles are congruent.
- [FOR RIGHT TRIANGLES] HL (Hypotenuse-Leg): If two right triangles have one pair of legs congruent and hypotenuses congruent, then the triangles are congruent.
If two triangles don't satisfy at least one of the criteria above, you cannot be confident that they are congruent.
Example A
\begin{align*}\overline{BC}\end{align*} is the perpendicular bisector of \begin{align*}\overline{AD}\end{align*}. Is \begin{align*}\triangle ABC \cong \triangle ADC\end{align*}?
Solution: If \begin{align*}\overline{BC}\end{align*} is the perpendicular bisector of \begin{align*}\overline{AD}\end{align*}, then \begin{align*}\overline{AC}\cong \overline{CD}\end{align*}. Also, \begin{align*}m\angle ACB=90^\circ\end{align*} and \begin{align*}m\angle DCB=90^\circ\end{align*}, so \begin{align*}\angle ACB \cong \angle DCB\end{align*}. You also know that \begin{align*}\overline{BC}\end{align*} is a side of both triangles, and is clearly congruent to itself (this is called the reflexive property).
The triangles are congruent by SAS. Note that even though these are right triangles, you would not use HL to show triangle congruence in this case since you are not given that the hypotenuses are congruent.
Example B
Using the information from Example A, if \begin{align*}m\angle A=50^\circ\end{align*}, what is \begin{align*}m\angle D\end{align*}?
Solution: \begin{align*}m\angle D=50^\circ\end{align*}. Since the triangles are congruent, all of their corresponding angles and sides must be congruent. \begin{align*}\angle A\end{align*} and \begin{align*}\angle D\end{align*} are corresponding angles, so \begin{align*}\angle A \cong \angle D\end{align*}.
Example C
Does one diagonal of a rectangle divide the rectangle into congruent triangles?
Solution: Recall that a rectangle is a quadrilateral with four right angles. The opposite sides of a rectangle are congruent.
There is more than enough information to show that \begin{align*}\triangle EFG \cong \triangle GHE\end{align*}.
- Method #1: The triangles have three pairs of congruent sides, so they are congruent by SSS.
- Method #2: The triangles have two pairs of congruent sides and congruent included angles, so they are congruent by SAS.
- Method #3: The triangles are right triangles with congruent hypotenuses and a pair of congruent legs, so they are congruent by HL.
Concept Problem Revisited
Max constructs a triangle in Geogebra. He tells Alicia that his triangle has a \begin{align*}42^\circ\end{align*} angle, a side of length 12, and a side of length 8. If Max also told Alicia that the angle was in between the two sides, then she would be able to construct a triangle that must be congruent due to SAS. If the angle is not between the two sides, she cannot be confident that her triangle is congruent because SSA is not a criterion for triangle congruence. Because Max did not state where the angle was in relation to the sides, Alicia cannot create a triangle that must be congruent to Max's triangle.
Vocabulary
AAS (Angle-Angle-Side): If two triangles have two pairs of congruent angles and a non-common side of the angles in one triangle is congruent to the corresponding side in the other triangle, then the triangles are congruent.
ASA (Angle-Side-Angle): If two triangles have two pairs of congruent angles and the common side of the angles in one triangle is congruent to the corresponding side in the other triangle, then the triangles are congruent.
SAS (Side-Angle-Side): If two triangles have two pairs of congruent sides and the included angle in one triangle is congruent to the included angle in the other triangle, then the triangles are congruent.
SSS (Side-Side-Side): If two triangles have three pairs of congruent sides, then the triangles are congruent.
HL (Hypotenuse-Leg): If two right triangles have one pair of legs congruent and hypotenuses congruent, then the triangles are congruent.
Rigid transformations are transformations that preserve distance and angles. The rigid transformations are reflections, rotations, and translations.
Two figures are congruent if a sequence of rigid transformations will carry one figure to the other. Congruent figures will always have corresponding angles and sides that are congruent as well.
The perpendicular bisector of a segment is a line that intersects the segment at its midpoint and meets the segment at a right angle.
The midpoint is a point directly in between the two endpoints of the segment. It divides the segment into two congruent segments.
The reflexive property states that an object or quantity is equal to itself.
Guided Practice
For each pair of triangles, tell whether the given information is enough to show that the triangles are congruent. If the triangles are congruent, state the criterion that you used to determine the congruence and write a congruency statement. Note that the figures are not necessarily drawn to scale!
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3. \begin{align*}G\end{align*} is the midpoint of \begin{align*}\overline{EH}\end{align*}.
Answers:
- Notice that besides the one pair of congruent sides and the one pair of congruent angles, \begin{align*}\overline{AC}\cong \overline{CA}\end{align*}. \begin{align*}\triangle ACB \cong \triangle CAD \end{align*} by SAS.
- The congruent sides are not corresponding in the same way that the congruent angles are corresponding. The given information for \begin{align*}\triangle ACB\end{align*} is SAS while the given information for \begin{align*}\triangle CAD\end{align*} is SSA. The triangles are not necessarily congruent.
- Because \begin{align*}G\end{align*} is the midpoint of \begin{align*}\overline{EH}\end{align*}, \begin{align*}\overline{EG}\cong \overline{GH}\end{align*}. You also know that \begin{align*}\angle EGF \cong \angle HGI\end{align*} because they are vertical angles. \begin{align*}\triangle EGF \cong \triangle HGI\end{align*} by ASA.
Practice
1. List the five criteria for triangle congruence and draw a picture that demonstrates each.
2. Given two triangles, do you always need at least three pieces of information about each triangle in order to be able to state that the triangles are congruent?
For each pair of triangles, tell whether the given information is enough to show that the triangles are congruent. If the triangles are congruent, state the criterion that you used to determine the congruence and write a congruency statement.
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For 9-11, state whether the given information about a hidden triangle would be enough for you to construct a triangle that must be congruent to the hidden triangle. Explain your answer.
9. \begin{align*}\triangle ABC\end{align*} with \begin{align*}m\angle A=72^\circ\end{align*}, \begin{align*}AB=6 \ cm\end{align*}, \begin{align*}BC=8 \ cm\end{align*}.
10. \begin{align*}\triangle ABC\end{align*} with \begin{align*}m\angle A=90^\circ\end{align*}, \begin{align*}AB=4 \ cm\end{align*}, \begin{align*}BC=5 \ cm\end{align*}.
11. \begin{align*}\triangle ABC\end{align*} with \begin{align*}m\angle A=72^\circ\end{align*}, \begin{align*}AB=6 \ cm\end{align*}, \begin{align*}AC=8 \ cm\end{align*}.
12. Recall that a square is a quadrilateral with four right angles and four congruent sides. Show and explain why a diagonal of a square divides the square into two congruent triangles.
13. Show and explain using a different criterion for triangle congruence why a diagonal of a square divides the square into two congruent triangles.
14. Recall that a kite is a quadrilateral with two pairs of adjacent, congruent sides. Will one of the diagonals of a kite divide the kite into two congruent triangles? Show and explain your answer.
15. In the picture below, \begin{align*}G\end{align*} is the midpoint of both \begin{align*}\overline{EH}\end{align*} and \begin{align*}\overline{FI}\end{align*}. Explain why \begin{align*}\overline{FH}\cong \overline{IE}\end{align*} and \begin{align*}\overline{FE}\cong \overline{HI}\end{align*}.
16. Explain why AAA is not a criterion for triangle congruence.