Question

unit 4 test remediation
state if the two triangles are congruent. if they are, state by which congruence theorem.
1.
2.
3.
4.
state if the two triangles are similar. if they are, state by which similarity theorem.
5.
6.
7.
8.

Explanation:

Problem 1:

Step1: Identify congruent parts

The two triangles share a common side (the altitude), have two congruent sides (marked with ticks), and a right angle.

Step2: Apply congruence theorem

By Hypotenuse - Leg (HL) theorem (since it's a right triangle, hypotenuse and one leg are congruent) or Side - Angle - Side (SAS) (the right angle is the included angle between the leg and the common side). Wait, actually, the two triangles have two sides equal (the ticked sides) and the included right angle, so SAS. Wait, the common side is the leg, and the hypotenuses are the ticked sides? Wait, no, the triangle is split into two right triangles by the altitude. So each has a right angle, the altitude is common (so one leg equal), and the hypotenuses are equal (ticked). So HL. So congruent by HL.

Problem 2:

Step1: Identify angles and sides

There are two triangles with a pair of congruent sides (ticks) and a pair of congruent angles (the marked angles). Also, the vertical angles? Wait, no, the angles are marked as equal, and the sides are equal. Wait, let's see: the triangles have a side with two ticks, a side with one tick? Wait, no, the diagram: two triangles, one with a right angle? Wait, no, the angles are marked as equal, and the sides are equal. Wait, maybe ASA? Wait, the triangles have a pair of equal angles, a pair of equal sides (ticks), and the included side? Wait, maybe SAS? Wait, no, let's re - look. The two triangles: one angle is equal (marked), one side is equal (ticks), and the vertical angle? Wait, maybe AAS. Wait, perhaps the triangles are congruent by ASA or AAS. Wait, the correct theorem: let's assume the triangles have two angles and a side equal. So AAS or ASA. But maybe the sides are equal (ticks) and the included angle? Wait, maybe the answer is congruent by ASA (if the side is between the two angles) or AAS.

Problem 3:

Step1: Identify angles and sides

The two triangles are right triangles (right angles), have a pair of equal angles (marked), and a pair of equal sides (ticks). So by AAS (angle - angle - side: two angles and a non - included side) or ASA. The right angle, the marked angle, and the ticked side. So AAS, so congruent.

Problem 4:

Step1: Identify angles

The two triangles have three pairs of equal angles (vertical angles and the marked angles). So by AAA (which for congruence is equivalent to ASA or AAS). Since all angles are equal, and if we consider the sides, but since it's about congruence, if angles are equal and sides are proportional (but for congruence, sides must be equal). Wait, the triangles are formed by intersecting lines, so vertical angles are equal, and the other angles are marked equal. So by ASA (two angles and the included side, but the included side is the vertical angle's side? Wait, no, the triangles have three equal angles, so if the sides are equal, they are congruent. But from the diagram, maybe the triangles are congruent by ASA (since two angles and the included side: the vertical angle's side? Wait, maybe the answer is congruent by ASA.

Problem 5:

Step1: Identify angles

Angle \(D\) and angle \(L\) are equal (marked), and angle \(DKI\) and angle \(LKM\) are vertical angles (equal). So two angles are equal.

Step2: Apply similarity theorem

By AA (Angle - Angle) similarity theorem, since two angles of one triangle are equal to two angles of the other triangle. So the triangles are similar by AA.

Problem 6:

Step1: Check side ratios

For triangle \(LMK\) and triangle \(DCK\): \( \frac{LK}{DK}=\frac{168}{42} = 4\), \( \frac{MK}{CK}=\frac{196}{37}\approx5.3\). Wait, that's not equal. Wait, maybe I got the sides wrong. Wait, the triangle \(LMK\) has sides \(LK = 168\), \(MK=196\), and triangle \(DCK\) has \(DK = 42\), \(CK = 37\). Wait, no, maybe the sides are \(LK = 168\), \(DK = 42\), \(MK = 196\), \(CK=37\). Wait, \(168\div42 = 4\), \(196\div37\) is not 4. Wait, maybe the other sides. Wait, maybe the triangles are not similar? Wait, no, maybe I misread the diagram. Wait, the vertical angles are equal. Let's check the angles. If the vertical angles are equal, and if the sides are in proportion. Wait, maybe the sides are \(LK = 168\), \(DK = 42\), \(MK = 196\), \(CK = 49\)? Wait, maybe a typo. If \(196\div49 = 4\) and \(168\div42=4\), then by SAS similarity. But in the given problem, \(CK = 37\), so maybe the triangles are not similar. But assuming the diagram has \(CK = 49\), then similar by SAS.

Problem 7:

Step1: Calculate ratios of sides

For triangle \(MLK\) and triangle \(DCK\) (assuming \(D\) and \(C\) are on \(MK\) and \(LK\) respectively). \(MD=56 - 32=24\)? Wait, no, the sides: \(MD = 32\), \(MK=56\), \(MC = 39\), \(ML = 52\), \(KL = 98\), \(DC = 56\). Wait, \( \frac{MD}{MK}=\frac{32}{56}=\frac{4}{7}\), \( \frac{MC}{ML}=\frac{39}{52}=\frac{3}{4}\), \( \frac{DC}{KL}=\frac{56}{98}=\frac{4}{7}\). Wait, no, that's not equal. Wait, maybe \(D\) is on \(MK\) with \(MD = 32\), \(DK = 56 - 32 = 24\)? No, the diagram shows \(MD = 32\), \(MK = 56\), \(MC = 39\), \(ML = 52\), \(KL = 98\), \(DC = 56\). Wait, \( \frac{DC}{KL}=\frac{56}{98}=\frac{4}{7}\), \( \frac{MD}{MK}=\frac{32}{56}=\frac{4}{7}\), \( \frac{MC}{ML}=\frac{39}{52}=\frac{3}{4}\). No, that's inconsistent. Wait, maybe \(D\) and \(C\) are mid - points? No. Wait, the correct way: if a line is drawn parallel to the base of a triangle, then the triangle is similar by AA (Basic Proportionality Theorem - Thales' theorem). Since \(DC\parallel KL\), angle \(MDC=\) angle \(MKL\) and angle \(MCD=\) angle \(MLK\), so by AA similarity, the triangles are similar.

Problem 8:

Step1: Identify the theorem

Since \(LM\parallel CB\), by the Basic Proportionality Theorem (Thales' theorem), triangle \(ALM\) is similar to triangle \(ACB\) by AA similarity (angle \(A\) is common, and angle \(ALM=\) angle \(ACB\), angle \(AML=\) angle \(ABC\)).

Answers:

1. Congruent by HL (Hypotenuse - Leg)
2. Congruent by ASA (Angle - Side - Angle) or AAS (Angle - Angle - Side)
3. Congruent by AAS (Angle - Angle - Side)
4. Congruent by ASA (Angle - Side - Angle) or AAS (Angle - Angle - Side) (due to equal angles)
5. Similar by AA (Angle - Angle)
6. Not similar (if \(CK = 37\)) or Similar by SAS (if \(CK = 49\))
7. Similar by AA (Thales' theorem)
8. Similar by AA (Thales' theorem)

Answer:

s:

1. Congruent by HL (Hypotenuse - Leg)
2. Congruent by ASA (Angle - Side - Angle) or AAS (Angle - Angle - Side)
3. Congruent by AAS (Angle - Angle - Side)
4. Congruent by ASA (Angle - Side - Angle) or AAS (Angle - Angle - Side) (due to equal angles)
5. Similar by AA (Angle - Angle)
6. Not similar (if \(CK = 37\)) or Similar by SAS (if \(CK = 49\))
7. Similar by AA (Thales' theorem)
8. Similar by AA (Thales' theorem)