Concept Explanation:
The AAS (Angle-Angle-Side) Postulate states that if two angles and a non-included side of one triangle are congruent to two angles and a corresponding non-included side of another triangle, then the triangles are congruent. This means their corresponding sides and angles are identical in measurement.
The AAS postulate helps us determine triangle congruency without knowing all three sides or angles. It is important that the side is not the one between the two angles (that would be the ASA postulate).
Diagram:
Here’s a diagram of two triangles that demonstrate the AAS postulate:
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\begin{array}{c}
\begin{tikzpicture}
\draw (0,0) -- (4,0) -- (2,3) -- cycle;
\draw (6,0) -- (10,0) -- (8,3) -- cycle;
\draw (0,0) node[left] {A} -- (4,0) node[right] {B} -- (2,3) node[above] {C} -- cycle;
\draw (6,0) node[left] {D} -- (10,0) node[right] {E} -- (8,3) node[above] {F} -- cycle;
\draw (2.7, 0) node[below] {6};
\draw (8.7, 0) node[below] {6};
\draw (2,1.2) node {40^\circ};
\draw (8,1.2) node {40^\circ};
\draw (3,2) node {70^\circ};
\draw (9,2) node {70^\circ};
\end{tikzpicture}
\end{array}
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In the diagram, angles ( \angle A ) and ( \angle C ) are congruent to ( \angle D ) and ( \angle F ), and the non-included sides ( AB ) and ( DE ) are congruent.
Hard Questions:
Q1: In triangles ABC and DEF, ( \angle A = 50^\circ ), ( \angle B = 70^\circ ), and side ( AB = 8 ) cm. In triangle DEF, ( \angle D = 50^\circ ), ( \angle E = 70^\circ ), and side ( DE = 8 ) cm. Are the triangles congruent? If so, by which postulate?
Step-by-Step Solution:
We know two angles in triangle ABC are congruent to two angles in triangle DEF, and the non-included sides are congruent. By the AAS postulate, the triangles are congruent.
Answer:
- The triangles are congruent by the AAS postulate.
Q2: Triangle ABC has ( \angle A = 40^\circ ), ( \angle B = 65^\circ ), and side ( AB = 5 ) cm. Triangle DEF has ( \angle D = 40^\circ ), ( \angle E = 65^\circ ), and side ( DE = 5 ) cm. Find if the triangles are congruent and calculate the missing angles.
Step-by-Step Solution:
By the AAS postulate, the triangles are congruent because two angles and a non-included side are congruent. The missing angle ( C ) can be calculated as:
- ( \angle C = 180^\circ – 40^\circ – 65^\circ = 75^\circ )
Similarly, ( \angle F = 75^\circ ).
Answer:
- The triangles are congruent by the AAS postulate, and the missing angle is ( 75^\circ ).
Q3: Triangle XYZ has ( \angle X = 60^\circ ), ( \angle Y = 50^\circ ), and side ( XY = 7 ) units. Triangle PQR has ( \angle P = 60^\circ ), ( \angle Q = 50^\circ ), and side ( PQ = 7 ) units. Are the triangles congruent by the AAS postulate?
Step-by-Step Solution:
Since two angles and the non-included side in triangles XYZ and PQR are congruent, the triangles are congruent by the AAS postulate.
Answer:
- Yes, the triangles are congruent by the AAS postulate.