If ΔABC ≅ ΔXYZ and ∠BAC = 55°, then ∠ZXY = ?
55°
Understanding Triangle Congruence
The question deals with the concept of triangle congruence. When we say that two triangles are congruent, it means that they are exactly the same in size and shape. If you can place one triangle perfectly on top of the other so they match up exactly, they are congruent.
Mathematically, the congruence of two triangles ($\Delta$ABC $\cong$ $\Delta$XYZ) implies that all three corresponding sides are equal in length, and all three corresponding angles are equal in measure.
Corresponding Parts of Congruent Triangles
The order of the vertices in the congruence statement is very important. When we write $\Delta$ABC $\cong$ $\Delta$XYZ, it establishes a specific correspondence between the vertices of the two triangles:
- Vertex A corresponds to Vertex X
- Vertex B corresponds to Vertex Y
- Vertex C corresponds to Vertex Z
This vertex correspondence leads to the equality of corresponding sides and angles:
| Corresponding Sides | Corresponding Angles |
|---|---|
| AB = XY | $\angle$A = $\angle$X (or $\angle$BAC = $\angle$YXZ) |
| BC = YZ | $\angle$B = $\angle$Y (or $\angle$ABC = $\angle$XYZ) |
| AC = XZ | $\angle$C = $\angle$Z (or $\angle$BCA = $\angle$XZY) |
This principle is often summarized as CPCTC: Corresponding Parts of Congruent Triangles are Congruent (or Equal).
Solving the Angle Problem: Finding ∠ZXY
We are given that $\Delta$ABC $\cong$ $\Delta$XYZ and $\angle$BAC = 55°.
We need to find the measure of $\angle$ZXY.
Looking at the correspondence from the congruence statement $\Delta$ABC $\cong$ $\Delta$XYZ:
- The first vertex of $\Delta$ABC is A.
- The first vertex of $\Delta$XYZ is X.
This tells us that angle A in $\Delta$ABC corresponds to angle X in $\Delta$XYZ. The angle $\angle$BAC is the angle at vertex A in $\Delta$ABC. The angle $\angle$ZXY (which is the same as $\angle$YXZ) is the angle at vertex X in $\Delta$XYZ.
According to the property of congruent triangles, corresponding angles are equal.
Therefore, $\angle$BAC = $\angle$ZXY.
Since we are given $\angle$BAC = 55°, we can conclude that:
$\angle$ZXY = 55°.
Comparing with Options
Let's check the given options:
- 67.5°
- 55°
- 135°
- 65°
Our calculated value for $\angle$ZXY is 55°, which matches option 2.
Revision Table: Triangle Congruence Basics
| Concept | Description | Key takeaway for this problem |
|---|---|---|
| Triangle Congruence ($\cong$) | Two triangles are congruent if they have the same size and shape. | $\Delta$ABC $\cong$ $\Delta$XYZ means all corresponding parts are equal. |
| Corresponding Parts | Sides and angles that match up when two congruent shapes are superimposed. Identified by the order of vertices in the congruence statement. | A corresponds to X, B to Y, C to Z. $\angle$A corresponds to $\angle$X. |
| CPCTC | Corresponding Parts of Congruent Triangles are Congruent (or Equal). | Used to state that $\angle$BAC = $\angle$ZXY because they are corresponding angles. |
Additional Information: Properties of Congruent Triangles
Understanding triangle congruence is fundamental in geometry. Beyond the angles and sides being equal, congruence is transitive (if $\Delta$1 $\cong$ $\Delta$2 and $\Delta$2 $\cong$ $\Delta$3, then $\Delta$1 $\cong$ $\Delta$3) and symmetric (if $\Delta$1 $\cong$ $\Delta$2, then $\Delta$2 $\cong$ $\Delta$1). There are also several congruence postulates (SSS, SAS, ASA, AAS, HL) that allow us to prove triangles are congruent without checking all six parts.
- SSS (Side-Side-Side): If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
- SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
- ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.
- AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are equal to two angles and the corresponding non-included side of another triangle, the triangles are congruent.
- HL (Hypotenuse-Leg): If the hypotenuse and a leg of a right triangle are equal to the hypotenuse and a leg of another right triangle, the triangles are congruent.
These postulates are the tools used to establish congruence, which then allows us to use the CPCTC property to find unknown side lengths or angle measures, as demonstrated in this problem.
Important Questions from Triangles, Congruence and Similarity
Angle between the internal bisectors of two angles ∠B and ∠C of a ΔABC is 132°, then the value of ∠A is
In ΔPQR, PQ = PR and S is a point on QR such that ∠PSQ = 96° + ∠QPS and ∠QPR = 132°. What is the measure of ∠PSR?
In Δ ABC, ∠A = 50°. If the bisectors of the angle B and angle C, meet at a point O, then ∠BOC is equal to:
Triangle ABC is right angled at B. BD is an altitude intersecting AC at D. If AC = 9 cm and CD = 3 cm. then find the measure of AB (in cm).
The base and altitude of an isosceles triangle are 10 cm and 12 cm respectively. Then the length of each equal side is:
