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Questions and Answers
Two triangles, (\triangle ABC) and (\triangle XYZ), have (AB = XY) and (BC = YZ). What additional piece of information is needed to prove that (\triangle ABC \cong \triangle XYZ) using the SAS congruence postulate?
Two triangles, (\triangle ABC) and (\triangle XYZ), have (AB = XY) and (BC = YZ). What additional piece of information is needed to prove that (\triangle ABC \cong \triangle XYZ) using the SAS congruence postulate?
(\angle B \cong \angle Y)
Given (\triangle DEF) where (\angle D = 30^{\circ}) and (\angle E = 70^{\circ}), what is the measure of the exterior angle at vertex F?
Given (\triangle DEF) where (\angle D = 30^{\circ}) and (\angle E = 70^{\circ}), what is the measure of the exterior angle at vertex F?
100 degrees
In right triangles (\triangle PQR) and (\triangle LMN), (\angle Q) and (\angle M) are right angles, (PR \cong LN). What additional congruence is sufficient to prove (\triangle PQR \cong \triangle LMN) using the HL Congruence Theorem?
In right triangles (\triangle PQR) and (\triangle LMN), (\angle Q) and (\angle M) are right angles, (PR \cong LN). What additional congruence is sufficient to prove (\triangle PQR \cong \triangle LMN) using the HL Congruence Theorem?
Either (PQ \cong LM) or (QR \cong MN)
Suppose you have two triangles, (\triangle ABC) and (\triangle DEF), such that (\angle A \cong \angle D) and (\angle B \cong \angle E). What additional information is needed to prove (\triangle ABC \cong \triangle DEF) using the AAS congruence postulate?
Suppose you have two triangles, (\triangle ABC) and (\triangle DEF), such that (\angle A \cong \angle D) and (\angle B \cong \angle E). What additional information is needed to prove (\triangle ABC \cong \triangle DEF) using the AAS congruence postulate?
If (\triangle ABC \cong \triangle DEF) and (AB = 5), (BC = 7), and (AC = 10), what is the length of (DF)? Explain your answer.
If (\triangle ABC \cong \triangle DEF) and (AB = 5), (BC = 7), and (AC = 10), what is the length of (DF)? Explain your answer.
(\overline{AD}) bisects (\angle BAC) and (\overline{AD} \perp \overline{BC}) at point D. If (\triangle ABD \cong \triangle ACD), state one conclusion that can be justified using CPCTC.
(\overline{AD}) bisects (\angle BAC) and (\overline{AD} \perp \overline{BC}) at point D. If (\triangle ABD \cong \triangle ACD), state one conclusion that can be justified using CPCTC.
Given two overlapping triangles, (\triangle ACE) and (\triangle DBF), with (AC \cong DB) and (CE \cong BF). What additional information is needed to prove (\triangle ACE \cong \triangle DBF) using the SSS congruence postulate?
Given two overlapping triangles, (\triangle ACE) and (\triangle DBF), with (AC \cong DB) and (CE \cong BF). What additional information is needed to prove (\triangle ACE \cong \triangle DBF) using the SSS congruence postulate?
In (\triangle PQR), (\angle P = x), (\angle Q = 2x), and (\angle R = 3x). Find the measure of each angle and classify the triangle by its angles.
In (\triangle PQR), (\angle P = x), (\angle Q = 2x), and (\angle R = 3x). Find the measure of each angle and classify the triangle by its angles.
Flashcards
Congruent Triangles
Congruent Triangles
Triangles with the same size and shape. All corresponding sides and angles are congruent.
SSS Congruence
SSS Congruence
If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
SAS Congruence
SAS Congruence
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
ASA Congruence
ASA Congruence
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AAS Congruence
AAS Congruence
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HL Congruence
HL Congruence
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CPCTC
CPCTC
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Triangle Angle-Sum Theorem
Triangle Angle-Sum Theorem
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Study Notes
- Congruent triangles are triangles that have the same size and shape.
- Corresponding sides and corresponding angles are congruent in congruent triangles
- CPCTC (Corresponding parts of congruent triangles are congruent)
- If ∆ABC ≅ ∆XYZ, then AB ≅ XY, BC ≅ YZ, CA ≅ ZX, ∠A ≅ ∠X, ∠B ≅ ∠Y, ∠C ≅ ∠Z.
Triangle Congruence Criteria
- These are shortcuts to prove triangle congruence without showing all sides and angles are congruent.
- Side-Side-Side (SSS) Congruence: If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.
- Side-Angle-Side (SAS) Congruence: Congruence if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle.
- Angle-Side-Angle (ASA) Congruence: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
- Angle-Angle-Side (AAS) Congruence: If two angles and a non-included side of one triangle are congruent to the corresponding two angles and side of another triangle, then the triangles are congruent.
- Hypotenuse-Leg (HL) Congruence: Applies only to right triangles; congruence if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle.
Proving Triangle Congruence
- Begin by stating the given information.
- Use congruence postulates (SSS, SAS, ASA, AAS, HL) to prove that triangles are congruent.
- Use CPCTC to prove corresponding parts of congruent triangles congruent.
- Use auxiliary lines if needed to prove triangle congruence.
- Ensure each statement is justified with a valid reason (given, definition, postulate, theorem).
Applying Congruence Criteria
- Triangle congruence can be used to prove other geometric relationships.
- Overlapping triangles often require separation or mental visualization to prove congruence.
Extending Congruence Criteria
- Angle-Side-Side (ASS) congruence does not exist
- SSA does not guarantee triangle congruence.
- Congruence might be proven in right triangles (HL) or when the angle is obtuse and the side opposite is longer than the adjacent side
Angle Relationships in Triangles
- Interior angles of a triangle sum to 180° by the Triangle Angle-Sum Theorem
- Exterior Angle Theorem: An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
- Third Angles Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.
Isosceles Triangles
- An isosceles triangle has at least two congruent sides.
- Base Angles Theorem: The angles opposite the congruent sides (base angles) are congruent.
- Converse of Base Angles Theorem: If two angles of a triangle are congruent, the sides opposite are congruent.
- The vertex angle is formed by the two congruent sides.
- The base is opposite the vertex angle.
- The altitude from the vertex angle to the base bisects the base and the vertex angle.
- Equilateral triangles are also equiangular, with all angles congruent and measuring 60°.
- If a triangle is equiangular, then it is also equilateral.
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Description
Congruent triangles have the same size and shape, with congruent corresponding sides and angles. The criteria (SSS, SAS, ASA, AAS) provide shortcuts to prove triangle congruence without needing to show all sides and angles are congruent.
