triangle congruence worksheet #3
ii. for each pair of triangles, (a) are they congruent? (b) if so, write the triangle congruence statement. (c) give the postulate (sss, sas, asa, aas, hl) that makes them congruent. (d) list any additional information needed to prove them congruent (vertical angles, reflexive property, etc.).

1.
a. ______
b. $\delta \text{___} \cong \delta \text{___}$
c. ______
d. ______

2.
a. ______
b. $\delta \text{___} \cong \delta \text{___}$
c. ______
d. ______

3. given: t is the midpoint of wr
a. ______
b. $\delta \text{___} \cong \delta \text{___}$
c. ______
d. ______

4.
a. ______
b. $\delta \text{___} \cong \delta \text{___}$
c. ______
d. ______

5. given: $\overrightarrow{ih}$ bisects $\angle wis$
a. ______
b. $\delta \text{___} \cong \delta \text{___}$
c. ______
d. ______

6.
a. ______
b. $\delta \text{___} \cong \delta \text{___}$
c. ______
d. ______

7.
a. ______
b. $\delta \text{___} \cong \delta \text{___}$
c. ______
d. ______

8.
a. ______
b. $\delta \text{___} \cong \delta \text{___}$
c. ______
d. ______

9.
a. ______
b. $\delta \text{___} \cong \delta \text{___}$
c. ______
d. ______

Question

triangle congruence worksheet #3
ii. for each pair of triangles, (a) are they congruent? (b) if so, write the triangle congruence statement. (c) give the postulate (sss, sas, asa, aas, hl) that makes them congruent. (d) list any additional information needed to prove them congruent (vertical angles, reflexive property, etc.).

1.
a. ______
b. $\delta \text{___} \cong \delta \text{___}$
c. ______
d. ______

2.
a. ______
b. $\delta \text{___} \cong \delta \text{___}$
c. ______
d. ______

3. given: t is the midpoint of wr
a. ______
b. $\delta \text{___} \cong \delta \text{___}$
c. ______
d. ______

4.
a. ______
b. $\delta \text{___} \cong \delta \text{___}$
c. ______
d. ______

5. given: $\overrightarrow{ih}$ bisects $\angle wis$
a. ______
b. $\delta \text{___} \cong \delta \text{___}$
c. ______
d. ______

6.
a. ______
b. $\delta \text{___} \cong \delta \text{___}$
c. ______
d. ______

7.
a. ______
b. $\delta \text{___} \cong \delta \text{___}$
c. ______
d. ______

8.
a. ______
b. $\delta \text{___} \cong \delta \text{___}$
c. ______
d. ______

9.
a. ______
b. $\delta \text{___} \cong \delta \text{___}$
c. ______
d. ______

Accepted Answer

{ "quality": "clear", "question_count": 9, "skills_matched": [ "direct_cot", "explain_then_answer", "tutor" ], "needs_drawing": false, "concepts_used": [ "Midpoint Definition" ], "new_concepts": [ "Triangle Congruence Postulates", "Reflexive Property of Congruence", "Vertical Angles Theorem" ], "current_concepts": [ "Midpoint Definition", "Triangle Congruence Postulates", "Reflexive Property of Congruence", "Vertical Angles Theorem" ] } ## Analyze Question 1 In Question 1, we are given two right triangles, $\Delta CDE$ and $\Delta ABE$. - Right angles are marked at $\angle C$ and $\angle A$, so $\angle C \cong \angle A$. - Segment $DE$ is marked with a single tick mark, and segment $BE$ is marked with a single tick mark, so $DE \cong BE$. - The vertical angles $\angle DEC$ and $\angle BEA$ are congruent by the Vertical Angles Theorem. - Thus, we have two angles and a non-included side congruent: $\angle C \cong \angle A$ (Angle), $\angle DEC \cong \angle BEA$ (Angle), and $DE \cong BE$ (Side). - Therefore, the triangles are congruent by the Angle-Angle-Side (AAS) postulate. - The congruence statement is $\Delta CDE \cong \Delta ABE$. - The additional information used is vertical angles. ## Analyze Question 2 In Question 2, we have triangles $\Delta OLE$ and $\Delta OVE$. - Segment $OL$ is marked with double tick marks, and segment $OV$ is marked with double tick marks, so $OL \cong OV$. - Segment $LE$ is marked with a single tick mark, and segment $VE$ is marked with a single tick mark, so $LE \cong VE$. - Segment $OE$ is shared by both triangles, so $OE \cong OE$ by the Reflexive Property. - Thus, all three corresponding sides are congruent. - Therefore, the triangles are congruent by the Side-Side-Side (SSS) postulate. - The congruence statement is $\Delta OLE \cong \Delta OVE$. - The additional information used is the reflexive property. ## Analyze Question 3 In Question 3, we are given that $T$ is the midpoint of $WR$. - Using the **Midpoint Definition**, this means $WT \cong RT$. - Segment $AT$ is marked with a single tick mark, and segment $ET$ is marked with a single tick mark, so $AT \cong ET$. - Angle $\angle W$ is marked with a single arc, and angle $\angle R$ is marked with a single arc, so $\angle W \cong \angle R$. - We have two sides and a non-included angle (SSA), which is not a valid congruence postulate. - Therefore, the triangles cannot be proven congruent with the given information. ## Analyze Question 4 In Question 4, we have parallel lines $l \parallel m$. - Segment $EG$ is marked with a single tick mark, and segment $FH$ is marked with a single tick mark, so $EG \cong FH$. - Segment $FG$ is shared by $\Delta EGF$ and $\Delta HGF$, but the triangles of interest are $\Delta EGF$ and $\Delta HFG$. - Let's look at $\Delta EGF$ and $\Delta HFG$. They share the side $GF$, so $GF \cong FG$ by the Reflexive Property. - Since $l \parallel m$, the alternate interior angles $\angle EGF$ and $\angle HFG$ are congruent. - We have side $EG \cong FH$, included angle $\angle EGF \cong \angle HFG$, and side $GF \cong FG$. - Therefore, the triangles are congruent by the Side-Angle-Side (SAS) postulate. - The congruence statement is $\Delta EGF \cong \Delta HFG$. - The additional information used is alternate interior angles and the reflexive property. ## Analyze Question 5 In Question 5, we are given that $\vec{IH}$ bisects $\angle WIS$. - By the definition of an angle bisector, $\angle WIH \cong \angle SIH$. - Segment $IH$ is perpendicular to $WS$, so $\angle IHW \cong \angle IHS = 90^\circ$. - Segment $IH$ is shared by both triangles, so $IH \cong IH$ by the Reflexive Property. - We have two angles and the included side congruent: $\angle WIH \cong \angle SIH$ (Angle), $IH \cong IH$ (Side), and $\angle IHW \cong \angle IHS$ (Angle). - Therefore, the triangles are congruent by the Angle-Side-Angle (ASA) postulate. - The congruence statement is $\Delta WIH \cong \Delta SIH$. - The additional information used is the reflexive property and the definition of perpendicular/bisector. ## Analyze Question 6 In Question 6, we have a rectangle-like figure with diagonal $LE$. - Angle $\angle ULE$ is marked with double arcs, and angle $\angle GLE$ is marked with double arcs, so $\angle ULE \cong \angle GLE$. - Angle $\angle UEL$ is marked with single arcs, and angle $\angle GEL$ is marked with single arcs, so $\angle UEL \cong \angle GEL$. - Segment $LE$ is shared by both triangles, so $LE \cong EL$ by the Reflexive Property. - We have two angles and the included side congruent. - Therefore, the triangles are congruent by the Angle-Side-Angle (ASA) postulate. - The congruence statement is $\Delta ULE \cong \Delta GLE$. - The additional information used is the reflexive property. ## Analyze Question 7 In Question 7, we have two parallel lines cut by two parallel transversals, forming a parallelogram $RSTU$. - Segment $RU$ is marked with double arrows, and segment $ST$ is marked with double arrows, indicating $RU \parallel ST$. - Segment $RS$ is marked with single arrows, and segment $UT$ is marked with single arrows, indicating $RS \parallel UT$. - The diagonal $RT$ is shared, so $RT \cong TR$ by the Reflexive Property. - Since $RS \parallel UT$, alternate interior angles $\angle SRT \cong \angle UTR$. - Since $RU \parallel ST$, alternate interior angles $\angle URT \cong \angle STR$. - We have two angles and the included side congruent. - Therefore, the triangles are congruent by the Angle-Side-Angle (ASA) postulate. - The congruence statement is $\Delta RST \cong \Delta TUR$. - The additional information used is alternate interior angles and the reflexive property. ## Analyze Question 8 In Question 8, we have a quadrilateral $VWYZ$ with diagonals. - Segment $VW$ is marked with a single tick mark, and segment $YZ$ is marked with a single tick mark, so $VW \cong YZ$. - Angle $\angle WVY$ is marked with a double arc, and angle $\angle ZYV$ is marked with a double arc, so $\angle WVY \cong \angle ZYV$. - Segment $VY$ is shared by $\Delta WVY$ and $\Delta ZYV$, so $VY \cong YV$ by the Reflexive Property. - We have side $VW \cong YZ$, included angle $\angle WVY \cong \angle ZYV$, and side $VY \cong YV$. - Therefore, the triangles are congruent by the Side-Angle-Side (SAS) postulate. - The congruence statement is $\Delta WVY \cong \Delta ZYV$. - The additional information used is the reflexive property. ## Analyze Question 9 In Question 9, we have two triangles meeting at vertex $A$. - Angle $\angle H$ is marked with a single arc, and angle $\angle P$ is marked with a single arc, so $\angle H \cong \angle P$. - Angle $\angle T$ is marked with a double arc, and angle $\angle M$ is marked with a double arc, so $\angle T \cong \angle M$. - The vertical angles $\angle HAT$ and $\angle PAM$ are congruent by the Vertical Angles Theorem. - However, no side lengths are given or marked as congruent. - Since we only have angle-angle-angle (AAA) similarity but no side information, we cannot prove congruence. - Therefore, the triangles cannot be proven congruent. ## Question 1 a. Yes b. $\Delta CDE \cong \Delta ABE$ c. AAS d. Vertical angles ## Question 2 a. Yes b. $\Delta OLE \cong \Delta OVE$ c. SSS d. Reflexive property ## Question 3 a. No b. $\Delta$ N/A $\cong \Delta$ N/A c. N/A d. Midpoint definition gives $WT \cong RT$, but SSA is not a valid congruence postulate ## Question 4 a. Yes b. $\Delta EGF \cong \Delta HFG$ c. SAS d. Alternate interior angles, Reflexive property ## Question 5 a. Yes b. $\Delta WIH \cong \Delta SIH$ c. ASA d. Reflexive property, Definition of angle bisector, Perpendicular lines form congruent right angles ## Question 6 a. Yes b. $\Delta ULE \cong \Delta GLE$ c. ASA d. Reflexive property ## Question 7 a. Yes b. $\Delta RST \cong \Delta TUR$ c. ASA d. Alternate interior angles, Reflexive property ## Question 8 a. Yes b. $\Delta WVY \cong \Delta ZYV$ c. SAS d. Reflexive property ## Question 9 a. No b. $\Delta$ N/A $\cong \Delta$ N/A c. N/A d. Vertical angles, but AAA is not a valid congruence postulate { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Triangle Congruence Postulates" ] }

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