4.5: Triangle Congruence using SAS, HL & ASA
Learning Objectives
- Understand and apply the SAS Congruence Postulate.
- Identify the distinct characteristics and properties of right triangles.
- Understand and apply the HL Congruence Theorem.
- Understand and apply the ASA Congruence Postulate.
SAS Congruence
One more way to show two triangles are congruent is by the SAS Congruence Postulate.
Side-Angle-Side (SAS) Triangle Congruence Postulate
If two sides and the included angle in one triangle are congruent to two sides and the included angle in another triangle, then the two triangles are congruent.
SAS stands for the __________________ - __________________ - __________________ Triangle Congruence Postulate.
The order of the letters is very significant. You must have the angles between the two sides for the SAS postulate to be valid.
Notice in the diagram above, the congruent angles are in between the pair of congruent sides. First, we see that the left sides (the first S in SAS) of the triangle are congruent because of the double tic marks: \begin{align*}\overline{DC} \cong \overline{OR}\end{align*}. Next moving around the triangle in a counter-clockwise direction, we see that the angles (A in SAS) are congruent because of the double arcs: \begin{align*}\angle C \cong \angle R\end{align*}. Finally, the bottom sides (next part in the counter-clockwise direction) of the triangle (the last S in SAS) are congruent because of the single tic mark: \begin{align*}\overline{CE} \cong \overline{RN}\end{align*}.
The congruent side-angle-side pairs in these two triangles satisfy the SAS Triangle Congruence Postulate.
In SAS, the angle must be _______________________________ the two sides.
Example 1
What information would you need to prove that these two triangles were congruent using the SAS postulate?
A. the measures of \begin{align*}\angle HJG\end{align*} and \begin{align*}\angle STR \end{align*}
B. the measures of \begin{align*}\angle HGJ\end{align*} and \begin{align*}\angle SRT\end{align*}
C. the measures of \begin{align*}\overline{HJ}\end{align*} and \begin{align*}\overline{ST}\end{align*}
D. the measures of sides \begin{align*}\overline{GJ}\end{align*} and \begin{align*}\overline{RT}\end{align*}
If you are to use the SAS postulate to establish congruence, you need to have the measures of two sides and the angle in between them for both triangles.
So far, you have one side and one angle. So, you must use the other side adjacent to the same angle. In \begin{align*}\Delta GHJ\end{align*}, that side is \begin{align*}\overline{HJ}\end{align*}. In triangle \begin{align*}\Delta RST\end{align*}, the corresponding side is \begin{align*}\overline{ST}\end{align*}.
So, the correct answer is C.
Reading Check:
1. What do the letters SAS (in the SAS Congruence Postulate) stand for?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. Where must the angle be (in relation to the sides) for the SAS Congruence Postulate to work?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
3. True/False: Two right triangles are congruent if the first triangle has legs that are 6 inches and 8 inches in length, and the second triangles has legs that are 8 inches and 6 inches in length. (Hint: draw a picture to help you!)
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\begin{align*}{\;}\end{align*}
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Right Triangles
So far, the congruence postulates we have examined work on any triangle you can imagine. As you know, there are a number of types of triangles:
- Acute triangles have all angles measuring less than \begin{align*}90^\circ\end{align*}.
- Obtuse triangles have one angle measuring between \begin{align*}90^\circ\end{align*} and \begin{align*}180^\circ\end{align*}.
- Equilateral triangles have congruent sides, and all angles measure \begin{align*}60^\circ\end{align*}.
- Right triangles have one angle measuring exactly \begin{align*}90^\circ\end{align*}.
In right triangles, the sides have special names. The two sides adjacent to (or next to) the right angle are called legs and the side opposite the right angle is called the hypotenuse.
In a right triangle, the side opposite the right angle is called the __________________________.
In a right triangle, the sides next to the right angle are called _____________________.
Example 2
Which side of right triangle \begin{align*}BCD\end{align*} is the hypotenuse?
Looking at \begin{align*}\Delta BCD\end{align*}, you can identify \begin{align*}\angle CBD\end{align*} as a right angle (remember the little square tells us the angle is a right angle).
By definition, the hypotenuse of a right triangle is opposite the right angle. So, side \begin{align*}\overline{CD}\end{align*} is the hypotenuse.
Reading Check:
1. How many right angles are in a right triangle?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. Which side of a right triangle is across from the right angle?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
3. Which side of a right triangle is next to the right angle?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
4. In your own words, describe an acute triangle.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
5. In your own words, describe an obtuse triangle.
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\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
HL Congruence
The “H” and “L” stand for hypotenuse and leg of right triangles.
HL Congruence Theorem
If the hypotenuse and leg in one right triangle are congruent to the hypotenuse and leg in another right triangle, then the two triangles are congruent.
HL stands for ___________________________ - __________________ Congruence Theorem.
The HL Congruence Theorem only works for _______________________ triangles.
Example 3
What information would you need to prove that these two triangles were congruent using the HL theorem?
A. the measures of sides \begin{align*}\overline{EF}\end{align*} and \begin{align*}\overline{MN}\end{align*}
B. the measures of sides \begin{align*}\overline{DF}\end{align*} and \begin{align*}\overline{LN}\end{align*}
C. the measures of angles \begin{align*}\angle DEF\end{align*} and \begin{align*}\angle LMN \end{align*}
D. the measures of angles \begin{align*}\angle DFE\end{align*} and \begin{align*}\angle LNM \end{align*}
Since these are right triangles, you only need one leg and the hypotenuse to prove congruence. Legs \begin{align*}\overline{DE}\end{align*} and \begin{align*}\overline{LM}\end{align*} are congruent, so you need to find the lengths of the hypotenuses. The hypotenuse of \begin{align*}\Delta DEF\end{align*} is \begin{align*}\overline{EF}\end{align*}. The hypotenuse of \begin{align*}\Delta LMN\end{align*} is \begin{align*}\overline{MN}\end{align*}. So, you need to find the measures of sides \begin{align*}\overline{EF}\end{align*} and \begin{align*}\overline{MN}\end{align*}.
The correct answer is A.
Reading Check:
1. What do the letters HL (in the HL Congruence Theorem) stand for?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. What other congruence postulate is the HL Congruence Theorem very similar to?
(Hint: think about the angle in between the hypotenuse and leg of a right triangle. What do these three parts spell?)
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\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
ASA Congruence
One of the other ways you can prove congruence between two triangles is the ASA Congruence Postulate. To use the ASA Postulate to show that two triangles are congruent, you must identify two angles and the included side (the side in between them).
Angle-Side-Angle (ASA) Congruence Postulate
If two angles and the included side in one triangle are congruent to two angles and the included side in another triangle, then the two triangles are congruent. ASA stands for the ___________________ - ___________________ - ___________________ Congruence Postulate.
In the ASA Postulate, you must use two angles and the side in ______________________ them.
Notice also that by picking two of the angles of the triangle, you have determined the measure of the third by the Triangle Sum Theorem. So, in reality, you have defined the whole triangle; you have identified all of the angles in the triangle, and by picking the length of one side, you defined the scale.
So, no matter what, if you have two angles and the side in between them, you have described the whole triangle.
The diagram above shows two congruent triangles using parts from the ASA Congruence Postulate. Moving counter-clockwise, the markings on the triangles show congruent matching left angles, then bottom sides, then right angles:
\begin{align*}\angle DCE & \cong \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\\ \overline{CE} & \cong \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\\ \angle CED & \cong \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
Example 4
What information would you need to prove that these two triangles are congruent using the ASA postulate?
A. the measures of the missing angles
B. the measures of sides \begin{align*}\overline{AB}\end{align*} and \begin{align*}\overline{BC}\end{align*}
C. the measures of sides \begin{align*}\overline{BC}\end{align*} and \begin{align*}\overline{EF}\end{align*}
D. the measures of sides \begin{align*}\overline{AC}\end{align*} and \begin{align*}\overline{DF}\end{align*}
If you are to use the ASA Postulate to prove congruence, you need to have two pairs of congruent angles and the included side, the side in between the pairs of congruent angles.
The side in between the two marked angles in \begin{align*}\Delta ABC\end{align*} is side \begin{align*}\overline{BC}\end{align*}. The side in between the two marked angles in \begin{align*}\Delta DEF\end{align*} is side \begin{align*}\overline{EF}\end{align*}. You would need the measures of sides \begin{align*}\overline{BC}\end{align*} and \begin{align*}\overline{EF}\end{align*} to prove congruence.
In total, you would then have the following:
\begin{align*}\angle ABC & \cong \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\\ \overline{BC} & \cong \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\\ \angle ACB & \cong \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}
The correct answer is C.
Reading Check:
1. What do the letters ASA (in the ASA Congruence Postulate) stand for?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. Why are the Third Angle Theorem and the ASA Congruence Postulate similar?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
3. In the diagram below, which triangle congruence postulate would you use to prove that the two triangles are congruent?
Image Attributions
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