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Questions and Answers
Which postulate can be used to prove two triangles are congruent if all three sides of one triangle are congruent to the corresponding three sides of another triangle?
Which postulate can be used to prove two triangles are congruent if all three sides of one triangle are congruent to the corresponding three sides of another triangle?
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, which postulate proves the two triangles are congruent?
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, which postulate proves the two triangles are congruent?
Two angles and the included side of one triangle are congruent to two angles and the included side of another triangle. Which postulate proves congruence?
Two angles and the included side of one triangle are congruent to two angles and the included side of another triangle. Which postulate proves congruence?
Two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle. What postulate proves congruence in this case?
Two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle. What postulate proves congruence in this case?
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What is the name for a triangle that can be divided into multiple smaller triangles?
What is the name for a triangle that can be divided into multiple smaller triangles?
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What property states that any segment or angle is congruent to itself?
What property states that any segment or angle is congruent to itself?
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If point M is the midpoint of segment AB, what can be concluded about segments AM and MB?
If point M is the midpoint of segment AB, what can be concluded about segments AM and MB?
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In a proof, it is stated that segment BD is congruent to segment BD. Which property justifies this statement?
In a proof, it is stated that segment BD is congruent to segment BD. Which property justifies this statement?
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Which postulate proves two triangles congruent if two angles and the included side of one triangle are congruent to the corresponding parts of another triangle?
Which postulate proves two triangles congruent if two angles and the included side of one triangle are congruent to the corresponding parts of another triangle?
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If $\overline{AB} \cong \overline{XY}$ and $\overline{BC} \cong \overline{YZ}$, what additional congruence is needed to prove $\triangle ABC \cong \triangle XYZ$ by SAS?
If $\overline{AB} \cong \overline{XY}$ and $\overline{BC} \cong \overline{YZ}$, what additional congruence is needed to prove $\triangle ABC \cong \triangle XYZ$ by SAS?
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What property justifies the statement that $\overline{PQ} \cong \overline{PQ}$?
What property justifies the statement that $\overline{PQ} \cong \overline{PQ}$?
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If $\angle ABC \cong \angle DEF$ and $\angle BCA \cong \angle EFD$, what additional congruence is needed to prove $\triangle ABC \cong \triangle DEF$ by AAS?
If $\angle ABC \cong \angle DEF$ and $\angle BCA \cong \angle EFD$, what additional congruence is needed to prove $\triangle ABC \cong \triangle DEF$ by AAS?
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Which of the following statements is true regarding an angle bisector?
Which of the following statements is true regarding an angle bisector?
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If point M is the midpoint of segment $\overline{AB}$, what conclusion can be drawn?
If point M is the midpoint of segment $\overline{AB}$, what conclusion can be drawn?
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What does CPCTC stand for?
What does CPCTC stand for?
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In $\triangle ABC$ and $\triangle DEF$, if $\overline{AB} \cong \overline{DE}$, $\overline{BC} \cong \overline{EF}$, and $\overline{AC} \cong \overline{DF}$, what postulate proves $\triangle ABC \cong \triangle DEF$?
In $\triangle ABC$ and $\triangle DEF$, if $\overline{AB} \cong \overline{DE}$, $\overline{BC} \cong \overline{EF}$, and $\overline{AC} \cong \overline{DF}$, what postulate proves $\triangle ABC \cong \triangle DEF$?
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Which term describes the intersection of two lines forming two pairs of congruent opposite angles?
Which term describes the intersection of two lines forming two pairs of congruent opposite angles?
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If a segment is divided into three congruent segments, it is said that the segment is:
If a segment is divided into three congruent segments, it is said that the segment is:
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Study Notes
Determining Congruent Triangles
- SSS Postulate (Side-Side-Side): If three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent.
- SAS Postulate (Side-Angle-Side): 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 Postulate (Angle-Side-Angle): 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.
- AAS Postulate (Angle-Angle-Side): If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the triangles are congruent.
Common Triangle Proofs
- Composite Triangle: A triangle that can be divided into two smaller triangles.
- Vertical Angles: When two lines intersect, they form two pairs of vertical angles, which are always congruent.
- Reflexive Property: Any segment or angle is congruent to itself.
- Midpoint: A midpoint divides a segment into two congruent segments.
Example Proofs
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Example 1: Prove triangle ABD is congruent to triangle CBD using the given information that AD is congruent to CD and B is the midpoint of AC.
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Statement: AD is congruent to CD
- Reason: Given
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Statement: B is the midpoint of AC
- Reason: Given
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Statement: AB is congruent to BC
- Reason: Definition of Midpoint
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Statement: BD is congruent to BD
- Reason: Reflexive Property
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Statement: Triangle ABD is congruent to triangle CBD
- Reason: SSS Postulate (Statements 1, 3, and 4)
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Statement: AD is congruent to CD
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Example 2: Prove triangle MRO is congruent to triangle PRO using the given information that segment RO is perpendicular to segment MP and segment MO is congruent to segment OP.
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Statement: MO is congruent to OP
- Reason: Given
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Statement: RO is perpendicular to MP
- Reason: Given
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Statement: Angle MOR is congruent to angle POR
- Reason: Perpendicular lines form right angles.
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Statement: RO is congruent to RO
- Reason: Reflexive Property
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Statement: Triangle MRO is congruent to triangle PRO
- Reason: SAS Postulate (Statements 1, 3, and 4)
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Statement: MO is congruent to OP
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Example 3: Prove triangle ABC is congruent to triangle ECD using the given information that angle one is congruent to angle 4 and segment AC is congruent to segment EC.
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Statement: Angle 1 is congruent to angle 4
- Reason: Given
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Statement: Segment AC is congruent to segment EC
- Reason: Given
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Statement: Angle 2 is congruent to angle 3
- Reason: Vertical angles are congruent.
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Statement: Triangle ABC is congruent to triangle ECD
- Reason: ASA Postulate (Statements 1, 2, and 3)
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Statement: Angle 1 is congruent to angle 4
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Example 4: Prove triangle ABE is congruent to triangle DCE using the given information that segment AB is congruent to segment CD, segment AE is congruent to segment DE, and angle 1 is congruent to angle 4.
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Statement: Segment AB is congruent to segment CD
- Reason: Given
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Statement: Segment AE is congruent to segment DE
- Reason: Given
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Statement: Angle 1 is congruent to angle 4
- Reason: Given
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Statement: Angle 2 is congruent to angle 3
- Reason: If two angles are congruent, then their supplementary angles are congruent.
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Statement: Triangle ABE is congruent to triangle DCE
- Reason: AAS Postulate (Statements 1, 3, and 4)
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Statement: Segment AB is congruent to segment CD
Proving Triangles Congruent
- SAS Postulate (Side-Angle-Side): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent.
- AAS Postulate (Angle-Angle-Side): If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, the triangles are congruent.
- SSS Postulate (Side-Side-Side): If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.
- ASA Postulate (Angle-Side-Angle): If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent.
Proving Triangles Congruent through Midpoints and Angle Bisectors
- Midpoint: A midpoint divides a segment into two congruent segments.
- Angle Bisector: An angle bisector is a ray that divides an angle into two congruent angles.
- Vertical Angles: Vertical angles are congruent.
Proving Triangles Congruent Through Trisectors
- Segment Trisector: A segment trisector divides a segment into three congruent segments.
Proving Triangles Congruent By Combining Strategies
- Segment Addition Property: To prove two segments are congruent, add a common segment to both sides of an equation that states two other segments are congruent.
- Definition of Congruent Segments: Segments are congruent if they have equal lengths.
- Definition of Congruent Angles: Angles are congruent if they have the same measure.
CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
- If two triangles are congruent, then their corresponding sides and angles are congruent.
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Description
Test your understanding of triangle congruence with this quiz covering the SSS, SAS, ASA, and AAS postulates. Explore the properties that prove triangles are congruent and test your knowledge of common triangle proofs. Perfect for geometry students looking to reinforce their learning.
