fill in the blank 5 points
state how the triangles are congruent using sss, sas, asa, aas, or hl. if they are not congruent, type not.
image of two triangles
type your answer...
14 fill in the blank 4 points
state how the triangles are congruent using sss, sas, asa, aas, or hl. if they are not congruent, type not.
image of a quadrilateral-like figure with triangles
type your answer...
15 fill in the blank 4 points
state how the triangles are congruent using sss, sas, asa, aas, or hl. if they are not congruent, type not.
image of two triangles
type your answer...

Question

Accepted Answer

### Problem 1 (Top Triangle Congruence)
# Explanation:
## Step1: Identify Right Angles
Triangles $ \triangle RE N $ (with right angle at $ E $) and $ \triangle PST $ (with right angle at $ S $) are right triangles.
## Step2: Identify Equal Sides
One leg of each right triangle is marked equal, and the hypotenuse of $ \triangle REN $ and one leg of $ \triangle PST $? Wait, no—wait, looking at the marks: $ RE $ and $ PS $? Wait, no, the first triangle (left) has right angle at $ E $, legs $ RE $ (vertical) and $ EN $ (horizontal), hypotenuse $ RN $. The second triangle (right) has right angle at $ S $, legs $ PS $ (horizontal) and $ ST $ (vertical), hypotenuse $ PT $. Wait, the marks: $ EN $ and $ PS $ are marked equal (horizontal legs), $ RN $ and $ PT $? No, wait, the horizontal leg of first triangle (EN) and horizontal leg of second (PS) are equal? Wait, no, the first triangle's horizontal side (EN) is marked equal to... Wait, maybe I missee. Wait, the first triangle: right angle at $ E $, so $ \angle E = 90^\circ $, $ EN $ and $ RE $ are legs, $ RN $ hypotenuse. The second triangle: right angle at $ S $, $ \angle S = 90^\circ $, $ PS $ and $ ST $ are legs, $ PT $ hypotenuse. The marks: $ EN $ (horizontal leg of first) and $ PS $ (horizontal leg of second) are equal? Wait, no, the first triangle's horizontal side (EN) is marked with two ticks, and the second's horizontal side (PS) is... Wait, maybe the hypotenuse of first (RN) and one leg of second? No, wait, the correct congruence: in right triangles, if a leg and hypotenuse are equal, it's HL. Wait, but maybe the legs: $ EN = PS $ (horizontal legs), $ \angle E = \angle S = 90^\circ $, and $ RE = ST $ (vertical legs)? Wait, the vertical leg of first (RE) and vertical leg of second (ST) are marked? Wait, the first triangle's vertical leg (RE) is unmarked, but the hypotenuse (RN) has one tick, and the second triangle's vertical leg (ST) has one tick? Wait, no, the first triangle: hypotenuse RN has one tick, vertical leg RE is unmarked, horizontal leg EN has two ticks. The second triangle: vertical leg ST has one tick, horizontal leg PS has two ticks, hypotenuse PT is unmarked. Wait, so $ EN = PS $ (two ticks), $ \angle E = \angle S = 90^\circ $, and $ RE = ST $? No, RE is unmarked, ST has one tick. Wait, maybe the hypotenuse RN and leg ST? No, that's not. Wait, maybe AAS? Wait, no, right triangles: if we have a leg and hypotenuse, HL. Wait, maybe I made a mistake. Wait, the correct answer: looking at the marks, $ EN = PS $ (horizontal legs, two ticks), $ \angle E = \angle S = 90^\circ $, and $ \angle N = \angle T $ (the acute angles, marked with one tick). So $ \angle E = \angle S = 90^\circ $, $ \angle N = \angle T $, and $ EN = PS $, so AAS. Wait, but AAS for right triangles? Or HL? Wait, no, AAS: two angles and a non-included side. So $ \angle E = \angle S = 90^\circ $, $ \angle N = \angle T $, and $ EN = PS $ (side between $ \angle E $ and $ \angle N $ in first triangle, and between $ \angle S $ and $ \angle T $ in second? No, $ EN $ is adjacent to $ \angle E $ and $ \angle N $, $ PS $ is adjacent to $ \angle S $ and $ \angle T $. So $ \angle E = \angle S $, $ \angle N = \angle T $, $ EN = PS $, so AAS. Wait, but maybe HL? Wait, no, HL requires hypotenuse and leg. Wait, maybe the hypotenuse RN and leg PT? No, marks: RN has one tick, PT is unmarked. Wait, maybe the correct congruence is AAS? Wait, no, let's re-express:

$ \triangle REN $ and $ \triangle TSP $ (wait, labels: R, E, N; T, S, P). $ \angle E = \angle S = 90^\circ $, $ \angle N = \angle T $ (marked), $ EN = PS $ (marked). So AAS (two angles and a non-included side). Wait, but in right triangles, AAS is valid. Alternatively, HL? No, HL needs hypotenuse and leg. Wait, maybe I'm wrong. Wait, the correct answer is AAS? Or HL? Wait, no, let's check again. The first triangle: right angle at E, so $ \angle E = 90^\circ $, sides: EN (horizontal, two ticks), RE (vertical, unmarked), RN (hypotenuse, one tick). The second triangle: right angle at S, $ \angle S = 90^\circ $, sides: PS (horizontal, two ticks), ST (vertical, one tick), PT (hypotenuse, unmarked). So $ EN = PS $ (two ticks), $ \angle E = \angle S = 90^\circ $, $ RN = ST $? No, RN has one tick, ST has one tick. Oh! Wait, RN (hypotenuse of first) has one tick, ST (leg of second) has one tick? No, ST is vertical leg of second, RN is hypotenuse of first. Wait, no, the vertical leg of first (RE) is unmarked, vertical leg of second (ST) has one tick. Hypotenuse of first (RN) has one tick, hypotenuse of second (PT) is unmarked. Wait, maybe the vertical leg of first (RE) and vertical leg of second (ST) are equal? No, RE is unmarked, ST has one tick. Wait, I think I messed up the labels. Let's re-express: Triangle 1: vertices R, E, N, right angle at E. So sides: RE (vertical), EN (horizontal), RN (hypotenuse). Triangle 2: vertices P, S, T, right angle at S. So sides: PS (horizontal), ST (vertical), PT (hypotenuse). The marks: EN (horizontal, two ticks) and PS (horizontal, two ticks) are equal. RN (hypotenuse, one tick) and ST (vertical, one tick) are equal? Wait, RN is hypotenuse (length $ \sqrt{RE^2 + EN^2} $), ST is leg (length ST). That can't be. Wait, no, maybe the vertical leg of first (RE) and vertical leg of second (ST) are equal (one tick each), and horizontal leg (EN and PS) are equal (two ticks each). Then, by SAS: $ RE = ST $, $ \angle E = \angle S = 90^\circ $, $ EN = PS $, so SAS. But SAS for right triangles is same as HL? No, HL is hypotenuse and leg. Wait, if $ RE = ST $ (leg), $ EN = PS $ (leg), and $ \angle E = \angle S = 90^\circ $, then SAS. But in right triangles, SAS is valid, and HL is for hypotenuse and leg. Wait, maybe the correct answer is AAS, SAS, or HL. Wait, I think the intended answer is AAS: $ \angle E = \angle S = 90^\circ $, $ \angle N = \angle T $ (marked), $ EN = PS $ (marked), so AAS.

## Step3: Determine Congruence Criterion
Since $ \angle E = \angle S = 90^\circ $, $ \angle N = \angle T $, and $ EN = PS $, the triangles are congruent by AAS. Wait, but maybe HL? No, HL needs hypotenuse and leg. Wait, maybe the hypotenuse RN and leg PT? No, marks don't show that. Alternatively, SAS: $ RE = ST $ (one tick), $ \angle E = \angle S = 90^\circ $, $ EN = PS $ (two ticks), so SAS. But I think the correct answer is AAS. Wait, no, let's check the angle marks: $ \angle N $ and $ \angle T $ are marked with one tick, so they are equal. $ \angle E $ and $ \angle S $ are right angles (equal). Then, the side between $ \angle E $ and $ \angle N $ is $ EN $, and between $ \angle S $ and $ \angle T $ is $ PS $, which are equal. So AAS (two angles and a non-included side). So the congruence is AAS.

# Answer: AAS

### Problem 2 (Middle Triangle Congruence: $ \triangle GHI $ and $ \triangle GJI $)
# Explanation:
## Step1: Identify Common Side
Triangles $ \triangle GHI $ and $ \triangle GJI $ share side $ GI $.
## Step2: Identify Equal Angles and Sides
$ \angle H = \angle J $ (marked with arcs), $ \angle HGI = \angle JGI $ (since $ GI $ is a common side and the angle is shared? Wait, no, the diagram shows $ GI $ as a diagonal, with $ \angle H $ and $ \angle J $ equal, and $ GH = GJ $? No, the sides: $ GH $ and $ GJ $ are equal? Wait, the triangles are $ \triangle GHI $ and $ \triangle GJI $, with $ GI $ common. $ \angle H = \angle J $ (marked), $ \angle HIG = \angle JIG $ (marked with ticks on the angle). So by AAS: $ \angle H = \angle J $, $ \angle HIG = \angle JIG $, $ GI = GI $ (common side). So AAS. Alternatively, SAS: if $ GH = GJ $, $ \angle H = \angle J $, $ HI = JI $? No, the marks: $ \angle H $ and $ \angle J $ are equal (arcs), $ \angle HIG $ and $ \angle JIG $ are equal (ticks), and $ GI $ is common. So AAS (two angles and a non-included side: $ \angle H, \angle HIG, GI $ and $ \angle J, \angle JIG, GI $).

## Step3: Determine Congruence Criterion
Since $ \angle H = \angle J $, $ \angle HIG = \angle JIG $, and $ GI = GI $ (common side), the triangles are congruent by AAS. Alternatively, ASA? Wait, $ \angle HIG = \angle JIG $, $ GI = GI $, $ \angle H = \angle J $: that's AAS (since the side is not between the two angles). Wait, ASA requires the side to be between the two angles. Here, $ GI $ is adjacent to $ \angle HIG $ and $ \angle JIG $, but $ \angle H $ and $ \angle J $ are opposite. So AAS.

# Answer: AAS

### Problem 3 (Bottom Triangle Congruence)
# Explanation:
## Step1: Identify Angles and Sides
First triangle: has a marked angle (top), two sides with one tick (left and right). Second triangle: has a right angle (bottom), two sides with one tick (left and top). Wait, the first triangle: isosceles? Wait, the first triangle has two sides with one tick (left and right), so it's isosceles with those sides equal. The second triangle has a right angle (bottom) and two sides with one tick (left and top). Wait, the angles: first triangle's top angle is marked, second's bottom angle is right angle. The sides: first triangle's two equal sides (one tick each) and second's two equal sides (one tick each). But the angles: first triangle's top angle and second's bottom angle (right angle) are not equal. Wait, the first triangle: let's say $ \triangle ABC $ with $ AB = AC $ (one tick each), $ \angle A $ marked. Second triangle: $ \triangle DEF $ with $ DE = DF $ (one tick each), $ \angle D = 90^\circ $. So $ AB = DE $ (one tick), $ AC = DF $ (one tick), but $ \angle A
eq \angle D $ (one is marked, one is right angle). So the triangles are not congruent? Wait, no, maybe I missee. Wait, the first triangle: two sides with one tick (left and right), top angle marked. Second triangle: two sides with one tick (left and top), bottom angle right angle. So the sides: first triangle's left and right sides (one tick) vs second's left and top sides (one tick). The angles: first's top angle vs second's bottom angle (right angle). So no congruence. So the answer is NOT.

## Step2: Check Congruence
Since the angles and sides don't match (one triangle has a marked angle, the other a right angle, and the equal sides are in different positions), the triangles are not congruent.

# Answer: NOT

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QUESTION IMAGE

Question

Explanation:

Problem 1 (Top Triangle Congruence)

Step1: Identify Right Angles

Step2: Identify Equal Sides

Step1: Identify Common Side

Step2: Identify Equal Angles and Sides

Step3: Determine Congruence Criterion

Step1: Identify Angles and Sides

Step2: Check Congruence

Answer:

Problem 2 (Middle Triangle Congruence: \( \triangle GHI \) and \( \triangle GJI \))