Proof Reasons Review Worksheet
Triangle Congruence Review
Proof Reasons Review Worksheet
Connecting geometric clues and diagram marks to valid mathematical conclusions.
Name:
Date:
A
Property & Direct Proof Conclusion Matching
Match each property (1–10) with its exact mathematical result (A–J).
1. Definition of Midpoint
2. Reflexive Property of Congruence
3. Definition of Vertical Angles
4. Vertical Angles are Congruent
5. Definition of Bisect
6. Definition of Perpendicular
7. All Right Angles are Congruent
8. Base Angles of Isosceles Δ are ≅
9. Definition of Equilateral Triangle
10. Alternate Interior Angles (Theorem)
A. If \(\overline{AB} \perp \overline{CD}\), then \(\angle 1\) and \(\angle 2\) are right angles.
B. If two parallel lines are cut by a transversal, then non-adjacent interior angles on opposite sides of the transversal are congruent.
C. Two angles whose sides form opposite rays across intersecting lines.
D. If \(\angle 1\) and \(\angle 2\) are vertical angles, then \(\angle 1 \cong \angle 2\).
E. Any geometric piece equals itself: \(\overline{AC} \cong \overline{AC}\) or \(\angle B \cong \angle B\).
F. If \(\triangle PQR\) is equilateral, then \(\overline{PQ} \cong \overline{QR} \cong \overline{RP}\).
G. If \(M\) is the midpoint of \(\overline{XY}\), then \(\overline{XM} \cong \overline{MY}\).
H. If \(\angle A\) and \(\angle B\) both measure \(90^\circ\), then \(\angle A \cong \angle B\).
I. In \(\triangle ABC\), if \(\overline{AB} \cong \overline{AC}\), then \(\angle B \cong \angle C\).
J. Ray \(\overrightarrow{BD}\) splits \(\angle ABC\) such that \(\angle ABD \cong \angle CBD\).
B
Visual Diagram Clue → Congruence Conclusion
Study each geometric diagram and given clue. Write the resulting Congruence Statement and the Exact Justifying Reason.
R T S midpoint
Given Condition: "Point \(T\) is the midpoint of \(\overline{RS}\)"
Congruence Statement:
Justifying Reason:
A B C D E
Given Condition: "\(\overline{AB}\) and \(\overline{CD}\) intersect at \(E\)"
Congruence Statement:
Justifying Reason:
X W Y Z
Given Condition: "\(\overline{WZ}\) is shared by \(\triangle XWZ\) & \(\triangle YWZ\)"
Congruence Statement:
Justifying Reason:
D A C B
Given Condition: "\(\overline{AD} \parallel \overline{BC}\) cut by transversal \(\overline{AC}\)"
Congruence Statement:
Justifying Reason:
Part C: Visual Criteria Classification Showdown
Triangle Congruence Review
Criteria Word Bank:
SSS SAS ASA AAS HL HA LA LL Not Enough Info
Examine each pair of triangles with its given tick marks, angle arcs, and shared features. Determine which criterion proves congruence.
Right triangles with congruent hypotenuses and one pair of congruent legs.
Criterion:
Right triangles with congruent hypotenuses and one pair of congruent acute angles.
Criterion:
Two outer sides marked congruent on each side, plus a shared central segment.
Criterion:
Right triangles with both pairs of corresponding legs congruent.
Criterion:
Right triangles with one pair of legs and one pair of acute angles congruent.
Criterion:
Two sides congruent and the angle directly between them (included) is congruent.
Criterion:
Two pairs of angles and the connecting side directly between them (included).
Criterion:
Two sides and a non-included acute angle are marked. Not a right triangle!
Criterion:
Proof Detective & Multi-Step Reasoning
Reasoning & Justification Mastery
D
Proof Detective: Find and Correct the Error
Read each flawed proof argument. Diagnose the mathematical error and write the correct justification.
Case 1
Student Argument: "Lines \(\overline{AB}\) and \(\overline{CD}\) intersect at \(X\). Therefore, \(\angle AXC \cong \angle BXD\) because of the Definition of Vertical Angles."
What is the error?
What is the correct reason?
Case 2
Student Argument: "Given \(\overline{XY} \perp \overline{AB}\), the next statement can immediately be \(\angle XYA \cong \angle XYB\) justified by Definition of Perpendicular."
What required step was skipped?
What reason proves the angles are \(\cong\)?
Case 3
Student Argument: "Two triangles have two congruent sides and one congruent angle (SSA). The student uses HL Theorem, but neither triangle contains a right angle."
Why can HL NOT be used here?
What must be true to use HL?
E
Multi-Step Diagram Reasoning
For the given figure, complete the sequence of statements and reasons that establish triangle congruence.
A B E C D Given: \(E\) is midpoint of \(\overline{AD}\) and \(\overline{BC}\).
1. Congruence: \(\overline{AE} \cong \overline{DE}\) Reason:
2. Congruence: \(\overline{BE} \cong \overline{CE}\) Reason:
3. Statement: \(\angle AEB\) and \(\angle DEC\) are vertical angles Reason:
4. Congruence: \(\angle AEB \cong \angle DEC\) Reason:
5. Conclusion: \(\triangle AEB \cong \triangle DEC\) Final Criterion: ______________
Proof Reasons Answer Key
Teacher Answer Key
Proof Reasons Answer Key
Complete solutions, justifying statements, and grading commentary.
Geometry Proof Review
Triangle Congruence Unit
A
Part A Solutions: Matching (1–10)
1. Definition of Midpoint → G
2. Reflexive Property of Congruence → E
3. Definition of Vertical Angles → C
4. Vertical Angles are Congruent → D
5. Definition of Bisect → J
6. Definition of Perpendicular → A
7. All Right Angles are Congruent → H
8. Base Angles of Isosceles Δ are ≅ → I
9. Definition of Equilateral Triangle → F
10. Alternate Interior Angles → B
B
Part B Solutions: Visual Diagram Clues
B1: Point \(T\) is midpoint of \(\overline{RS}\) \(\overline{RT} \cong \overline{TS}\) Definition of Midpoint
B2: \(\overline{AB}\) and \(\overline{CD}\) intersect at \(E\) \(\angle AEC \cong \angle DEB\) (or \(\angle AED \cong \angle CEB\)) Vertical Angles are Congruent
B3: Shared segment \(\overline{WZ}\) \(\overline{WZ} \cong \overline{WZ}\) Reflexive Property of Congruence
B4: \(\overline{AD} \parallel \overline{BC}\) transversal \(\overline{AC}\) \(\angle DAC \cong \angle BCA\) Alternate Interior Angles (Theorem)
C
Part C Solutions: Criteria Showdown (Cards 1–4)
1. Right Triangles (Hypotenuse + Leg): HL (Hypotenuse-Leg)
Applies only to right triangles. Hypotenuse and one leg correspond.
2. Right Triangles (Hypotenuse + Acute ∠): HA (Hypotenuse-Angle) (or AAS)
Right triangle shortcut equivalent to Angle-Angle-Side (AAS).
3. Shared Side Shape (2 outer pairs + center): SSS (Side-Side-Side)
The center shared segment is congruent by the Reflexive Property.
4. Right Triangles (Both pairs of legs): LL (Leg-Leg) (or SAS)
The right angle is the included angle between the two legs (SAS).
Proof Reasons Answer Key (Continued)
