triangle congruence
10. if two figures are similar, are the two figures also congruent? explain your answer in complete sentences.

for each pair of triangles, determine if the triangles are congruent (yes) or not (no). if they are congruent, then provide by which congruence theorem – sss, sas, asa, aas or hl.
11.
yes ■
no □
theorem:
12.
yes ■
no □
theorem:
13.
yes □
no ■
theorem:
14.
yes □
no ■
theorem:
15. using the image, the given information, and the statement to prove, add all appropriate markings and reasonings to the picture to complete a \picture proof.\
given: $angle m$ and $angle k$ are right angles; $overline{lm} cong overline{nk}$
prove: $angle mln cong angle knl$

Question

triangle congruence
10. if two figures are similar, are the two figures also congruent? explain your answer in complete sentences.

for each pair of triangles, determine if the triangles are congruent (yes) or not (no). if they are congruent, then provide by which congruence theorem – sss, sas, asa, aas or hl.
11.
yes ■
no □
theorem:
12.
yes ■
no □
theorem:
13.
yes □
no ■
theorem:
14.
yes □
no ■
theorem:
15. using the image, the given information, and the statement to prove, add all appropriate markings and reasonings to the picture to complete a \picture proof.\
given: $angle m$ and $angle k$ are right angles; $overline{lm} cong overline{nk}$
prove: $angle mln cong angle knl$

Accepted Answer

### Question 10
# Explanation:
## Step1: Recall definitions
Similar figures have the same shape (corresponding angles equal) and their corresponding sides are in proportion. Congruent figures have the same shape and same size (corresponding sides equal, which means the proportion of corresponding sides is $1$).
## Step2: Analyze the relationship
If two figures are similar, the ratio of corresponding sides is a scale factor $k$. For them to be congruent, $k = 1$ (so sides are equal). But similar figures can have $k
eq1$ (e.g., a small triangle and a larger similar triangle). So similarity doesn't imply congruence (though congruent figures are similar with $k = 1$).
# Answer:
No, two similar figures are not necessarily congruent. Similar figures have corresponding angles equal and corresponding sides in proportion (scale factor $k$). Congruent figures require the scale factor $k = 1$ (corresponding sides equal). Similar figures can have $k
eq1$ (e.g., a triangle and a larger similar triangle), so similarity does not guarantee congruence.

### Question 11
# Explanation:
## Step1: Identify given information
We have two right triangles. They share the hypotenuse, and two legs are marked equal (one pair of legs with one mark, another pair with two marks? Wait, looking at the diagram: both are right - angled, the hypotenuse is common ($\overline{AC}$ for $\triangle ABC$ and $\triangle ADC$, assuming the triangles are $\triangle ABC$ and $\triangle ADC$ with right angles at $B$ and $D$, $AB = AD$ (one mark) and $BC=DC$ (two marks)? Wait, actually, for right triangles, if we have $HL$ (Hypotenuse - Leg) or $SSS$ or $SAS$. Wait, the two right triangles: right angle, one leg equal, another leg equal, and hypotenuse common. So by $HL$ (Hypotenuse - Leg) or $SSS$ or $SAS$. Wait, the correct theorem here: for two right triangles, if the hypotenuse and a leg are equal, or two legs are equal. Wait, in the diagram, the two right triangles have the hypotenuse in common, and two legs (one pair with one tick, one pair with two ticks? Wait, maybe it's $HL$ or $SSS$. Wait, actually, the correct theorem here is $HL$ (Hypotenuse - Leg) or $SSS$. Wait, the two right triangles: right angle, hypotenuse is common, and one leg is equal (marked), another leg is equal (marked). So by $HL$ (since right angle, hypotenuse, and leg). Wait, the answer is Yes, and the theorem is $HL$ (or $SSS$ or $SAS$). Wait, let's re - examine: in right triangles, if we have two legs equal ($AB = AD$ and $BC = DC$) and hypotenuse $AC=AC$, then by $SSS$ (since $AB = AD$, $BC = DC$, $AC = AC$). Or if we consider the right angle, one leg, and hypotenuse: $HL$. So the triangles are congruent by $HL$ (or $SSS$).
## Step2: Apply congruence theorem
Since both are right - angled, hypotenuse is common, and the legs are equal (as per the markings), by the Hypotenuse - Leg ($HL$) congruence theorem (or $SSS$ as $AB = AD$, $BC = DC$, $AC = AC$) the triangles are congruent.
# Answer:
Yes, Theorem: $HL$ (or $SSS$)

### Question 12
# Explanation:
## Step1: Identify given information
We have two triangles with vertical angles (equal, since vertical angles are congruent), one pair of sides with one mark (equal), and one pair of angles (the non - vertical angles) marked equal. So we have $AAS$ (Angle - Angle - Side): vertical angles (equal), one angle (marked), and one side (marked).
## Step2: Apply congruence theorem
By $AAS$ (Angle - Angle - Side) congruence theorem, the two triangles are congruent. So the answer is Yes, and the theorem is $AAS$.
# Answer:
Yes, Theorem: $AAS$

### Question 13
# Explanation:
## Step1: Identify given information
We have two triangles. One triangle has two sides with two marks and one side with one mark, the other has two sides with two marks and one side with three marks? Wait, the first triangle: two sides with two ticks and one side with one tick. The second triangle: two sides with two ticks and one side with three ticks. So the sides are not equal (the side with one tick vs three ticks). So the triangles are not congruent.
## Step2: Conclusion
Since the corresponding sides are not equal (the lengths of the sides with different tick marks are different), the triangles are not congruent.
# Answer:
No, (no theorem applicable as sides are not equal)

### Question 14
# Explanation:
## Step1: Identify given information
We have two triangles with vertical angles (equal), one pair of sides with two marks (equal), and one pair of sides with one mark (equal). Wait, the vertical angles are equal, one side with two ticks, one side with one tick. So we have $SAS$ (Side - Angle - Side): two sides and the included angle (vertical angle) are equal. Wait, but the answer is marked as No. Wait, maybe the sides are not the included sides. Wait, the two triangles: the vertical angles are equal, but the sides with ticks are not the included sides. Wait, looking at the diagram: the two triangles have vertical angles, but the sides with two ticks and one tick are not the sides forming the vertical angle. So the congruence conditions (SSS, SAS, ASA, AAS, HL) are not met. So the triangles are not congruent.
## Step2: Conclusion
The triangles do not satisfy any of the congruence theorems (SSS, SAS, ASA, AAS, HL), so they are not congruent.
# Answer:
No, (no theorem applicable as the sides and angles do not satisfy any congruence condition)

### Question 15
# Explanation:
## Step1: Mark the given information on the diagram
- Mark $\angle M$ and $\angle K$ as right angles (draw a small square at $M$ and $K$).
- Mark $\overline{LM}\cong\overline{NK}$ (put the same number of ticks on $\overline{LM}$ and $\overline{NK}$).
- The common side is $\overline{LN}$ (so $\overline{LN}\cong\overline{LN}$ by reflexive property).
## Step2: Prove $\triangle MLN\cong\triangle KNL$
- We have $\angle M$ and $\angle K$ are right angles, so $\angle M=\angle K = 90^{\circ}$.
- $\overline{LM}\cong\overline{NK}$ (given).
- $\overline{LN}\cong\overline{LN}$ (reflexive property, common side).
- By $HL$ (Hypotenuse - Leg) congruence theorem, $\triangle MLN\cong\triangle KNL$.
- Since $\triangle MLN\cong\triangle KNL$, then $\angle MLN\cong\angle KNL$ (corresponding parts of congruent triangles are congruent - CPCTC).
# Answer:
- Mark $\angle M$ and $\angle K$ as right angles (square symbol).
- Mark $\overline{LM}$ and $\overline{NK}$ with the same tick marks.
- Reasoning:
    - $\angle M=\angle K = 90^{\circ}$ (given, right angles).
    - $\overline{LM}\cong\overline{NK}$ (given).
    - $\overline{LN}\cong\overline{LN}$ (reflexive property).
    - $\triangle MLN\cong\triangle KNL$ (HL).
    - $\angle MLN\cong\angle KNL$ (CPCTC).

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QUESTION IMAGE

Question

Explanation:

Question 10

Step1: Recall definitions

Step2: Analyze the relationship

Step1: Identify given information

Step2: Apply congruence theorem

Step1: Identify given information

Step2: Apply congruence theorem

Answer:

Question 11