Question

geometry, unit 5 – congruent triangles proof activity – part i name melanie chavez
for each problem, do the following:
a. show the given information in the diagram (using tick marks to show congruent sides and arcs to show congruent angles)
b. show any other congruent parts you notice (from vertical angles, sides shared in common, or alternate interior angles with parallel lines)
c. give the postulate or theorem that proves the triangles congruent (sss, sas, asa, aas, hl)
d. finally, fill in the blanks to complete the proof.
1.
given: $overline{bc} \cong \overline{dc}$ ; $overline{ac} \cong \overline{ec}$
prove: $\triangle bca \cong \triangle dce$

statements	reasons
2.	2. vertical $\angle$s theorem
3. $\triangle bca \cong \triangle dce$	3.

(diagram of triangles with vertices b, a, c and d, e, c intersecting at c)
2.
given: $overline{jk} \cong \overline{lk}$ ; $overline{jm} \cong \overline{lm}$
prove: $\triangle kjm \cong \triangle klm$

statements	reasons
2.	2. reflexive prop.
3.	3.

(diagram of triangles with vertices k, j, m and k, l, m with m as a common vertex)
3.
given: $\angle g \cong \angle i$ ; $overline{fh}$ bisects $\angle gfi$
prove: $\triangle gfh \cong \triangle ifh$

statements	reasons
2. $\angle gfh \cong \angle ifh$	2. def. of ____________
3.	3. reflexive prop.
4.	4.

(diagram of quadrilateral with vertices g, f, h, i and diagonal fh)

Explanation:

Step1: Solve Problem 1

Step 1a: Fill Statements 1

The given information is $\overline{BC} \cong \overline{DC}$ and $\overline{AC} \cong \overline{EC}$, so Statement 1 is $\overline{BC} \cong \overline{DC}$; $\overline{AC} \cong \overline{EC}$.

Step 1b: Fill Statements 2

By Vertical Angles Theorem, $\angle BCA \cong \angle DCE$, so Statement 2 is $\angle BCA \cong \angle DCE$.

Step 1c: Fill Reasons 3

We have two sides and the included angle congruent (SAS: Side - Angle - Side), so Reason 3 is SAS Postulate.

Step2: Solve Problem 2

Step 2a: Fill Statements 1

The given information is $\overline{JK} \cong \overline{LK}$ and $\overline{JM} \cong \overline{LM}$, so Statement 1 is $\overline{JK} \cong \overline{LK}$; $\overline{JM} \cong \overline{LM}$, and Reason 1 is Given.

Step 2b: Fill Statements 2

By Reflexive Property, the common side $\overline{KM} \cong \overline{KM}$, so Statement 2 is $\overline{KM} \cong \overline{KM}$.

Step 2c: Fill Statements 3 and Reasons 3

We have three sides congruent (SSS: Side - Side - Side), so Statement 3 is $\triangle KJM \cong \triangle KLM$ and Reason 3 is SSS Postulate.

Step3: Solve Problem 3

Step 3a: Fill Reasons 1

The given information is $\angle G \cong \angle I$ and $\overline{FH}$ bisects $\angle GFI$, so Reason 1 is Given.

Step 3b: Fill the blank in Reason 2

Since $\overline{FH}$ bisects $\angle GFI$, by Definition of Angle Bisector, $\angle GFH \cong \angle IFH$, so the blank is "Angle Bisector".

Step 3c: Fill Statements 3

By Reflexive Property, $\overline{FH} \cong \overline{FH}$, so Statement 3 is $\overline{FH} \cong \overline{FH}$.

Step 3d: Fill Statements 4 and Reasons 4

We have two angles and a non - included side congruent (AAS: Angle - Angle - Side), so Statement 4 is $\triangle GFH \cong \triangle IFH$ and Reason 4 is AAS Theorem.

Answer:

Problem 1:

• Statements 1: $\boldsymbol{\overline{BC} \cong \overline{DC}}$; $\boldsymbol{\overline{AC} \cong \overline{EC}}$
• Statements 2: $\boldsymbol{\angle BCA \cong \angle DCE}$
• Reasons 3: $\boldsymbol{SAS}$

Problem 2:

• Statements 1: $\boldsymbol{\overline{JK} \cong \overline{LK}}$; $\boldsymbol{\overline{JM} \cong \overline{LM}}$
• Reasons 1: $\boldsymbol{Given}$
• Statements 2: $\boldsymbol{\overline{KM} \cong \overline{KM}}$
• Statements 3: $\boldsymbol{\triangle KJM \cong \triangle KLM}$
• Reasons 3: $\boldsymbol{SSS}$

Problem 3:

• Reasons 1: $\boldsymbol{Given}$
• Reason 2: $\boldsymbol{Angle\ Bisector}$
• Statements 3: $\boldsymbol{\overline{FH} \cong \overline{FH}}$
• Statements 4: $\boldsymbol{\triangle GFH \cong \triangle IFH}$
• Reasons 4: $\boldsymbol{AAS}$