Geometric Twins · Lesson 4 of 10
Measuring Two Sides and the Included Angle
“Discover why an angle between two measured sides fixes a triangle, while three angles alone do not.”
• Explain why equal angles alone do not establish triangle congruence. • Identify the angle included between two given sides. • Construct a triangle from two sides and their included angle. • Use SAS with a correct vertex correspondence.
Do three angles fix the size?
A triangle has both sides and angles. Measuring its angles tells us how its corners open, but does it also tell us how large to draw it? Imagine enlarging a triangular drawing on a photocopier. Its sides become longer while its angles stay unchanged. The enlarged drawing will not fit the original exactly.
Triangles with angles 30°, 70°, and 80° can be drawn at different sizes. The angle sum is 180° in each, but none of the three angles specifies a length. Thus three equal corresponding angles, sometimes called AAA, do not guarantee congruence. We need enough information to fix size as well as shape.
Problem
One triangle has angles 30°, 70°, 80° and a side of 5 cm. An enlarged copy has the corresponding side 10 cm. Are they congruent?
- 1.Both have the same three angles, so the angle information agrees.
- 2.A corresponding side has changed from 5 cm to 10 cm. Exact superposition is impossible.
- 3.They are not congruent. AAA leaves the size undetermined.
The angle between the measured sides
There is a useful alternative to measuring all three sides. Choose two sides that meet at a vertex and measure the angle between them. That angle is called their included angle. It fixes how the two measured arms open, so their free endpoints are fixed as well. Joining those endpoints completes the triangle.
The angle at the common endpoint of two specified sides. Between AB and AC, the included angle is angle BAC, also written angle A.
To construct AB = 6 cm, AC = 5 cm, and angle BAC = 30°, first draw AB. At A make a ray forming 30° with AB. Mark C exactly 5 cm from A along this ray, then join C to B. The ray determines a direction and the 5 cm measurement determines a location on it. There is no freedom left to move C while preserving the data.
Two triangles are congruent if two corresponding sides and the angle included between those sides are equal. SAS stands for Side–Angle–Side.
Drawing the ray on the other side of AB produces a reflected triangle. It has the same shape and size and can match after a flip. This second orientation is not a different triangle size. As with SSS, congruence allows reflection.
Problem
AB = XY = 6 cm, AC = XZ = 5 cm, and angle BAC = angle YXZ = 30°. Prove congruence.
- 1.The two given sides in the first triangle meet at A; those in the second meet at X.
- 2.The given equal angles are exactly those included angles. Pair A ↔ X, B ↔ Y, and C ↔ Z.
- 3.Therefore triangle ABC ≅ triangle XYZ by SAS. In particular, BC = YZ and angles B and C match angles Y and Z.
Check the location of the angle
SAS is a precise condition, not a request for any two sides and any angle. Read the side names first and find their common endpoint. Then check that the given angle is at that endpoint. This quick check prevents a wrong proof, especially when the triangle is rotated or its labels are written in a different order.
Problem
PQ = LM = 7 cm and PR = LN = 5 cm. Which angle equality would establish SAS?
- 1.PQ and PR meet at P; LM and LN meet at L.
- 2.We need angle QPR = angle MLN, the angles at P and L.
- 3.With that equality, P ↔ L, Q ↔ M, and R ↔ N, giving PQR ≅ LMN by SAS. Angles at Q and M would not be included between these particular sides.
Two sides and an angle are not automatically SAS. The equal angle must lie between the two measured sides. Equal angles alone also leave the size free.
Quiz
Why does AAA fail to guarantee congruence?
Which is the included angle between AB and AC?
AB = DE, BC = EF. Which angle equality completes SAS?
What does SAS stand for?
With PQ = LM, PR = LN, and angle P = angle L, which statement follows?
Practice Problems
- Construct a triangle with AB = 6 cm, AC = 5 cm, and angle BAC = 30°. Explain what each step fixes.
- Draw a reflected copy of that triangle on the opposite side of AB and explain why it is congruent.
- Two triangles have angles 40°, 60°, and 80°. Explain why you still need information about size.
- For sides JK and KL, name the included angle using three letters.
- AB = DE = 7 cm, AC = DF = 5 cm, and angle A = angle D = 47°. Give the congruence statement and condition.
- Two sides of a triangle are 4 cm and 6 cm. Explain why an included angle of 30° and an included angle of 90° need not produce congruent triangles.
Key Takeaways
• Three matching angles alone do not fix a triangle’s size. • An included angle is at the vertex shared by the specified sides. • SAS fixes two side lengths and the opening between them. • A reflected construction is still congruent. • Check the angle’s position before using SAS.