Suppose line GH is congruent to line JK, line HE is congruent to line KL, and angle 1 is congruent to angle L. Can you prove that triangle GHI is congrunet to triangke JKL, abd if so, how?
A. You can use SAS to prove the triangles are congruent.
B. You can use SSS to prove the triangles are congruent.
C. Can use the definition of congruent triangles to prove the triangles are congruent.
D. There is not enough information to prove what the triangles are congruent.***
Triangle ABC has coordinates A(1, 4), B(5, 5), C(2, 2). Which of the following sets of coordinates are the vertices of a triangle that is congruent to triangle ABC?
A. (3, -1), (4, 5), (5, 7)
B. (-5, 1), (-1, 0), (-4, 3)
C. (0, 0), (-3, 3), (1, 4)***
D. (5, 3), (1, 2), (-2, -1)
Please help! I’m stuck!
The correct answer is C. (0, 0), (-3, 3), (1, 4).
For the first question, to prove that triangle GHI is congruent to triangle JKL, we can use the SAS (Side-Angle-Side) congruence theorem. Given that line GH is congruent to line JK (Sides GH and JK are congruent), line HE is congruent to line KL (Sides HE and KL are congruent), and angle 1 is congruent to angle L (Angle 1 and Angle L are congruent), we have enough information to establish congruence using SAS.
Therefore, the correct answer is: A. You can use SAS to prove the triangles are congruent.
For the second question, we can compare the side lengths and angles of triangle ABC with the given sets of coordinates to determine which set forms a congruent triangle.
Using the distance formula, we can find the lengths of the sides of triangle ABC:
AB = √((5-1)^2 + (5-4)^2) = √(16+1) = √17
BC = √((2-5)^2 + (2-5)^2) = √(9+9) = √18
AC = √((2-1)^2 + (2-4)^2) = √(1+4) = √5
Now let's check the answer choices:
A. (3, -1), (4, 5), (5, 7)
Using the distance formula, we find:
AB = √((4-3)^2 + (5+1)^2) = √(1+36) = √37
BC = √((5-4)^2 + (7-5)^2) = √(1+4) = √5
AC = √((5-3)^2 + (7+1)^2) = √(4+64) = √68
Since the side lengths of the new triangle are not congruent to those of triangle ABC, this set of coordinates does not form a congruent triangle.
B. (-5, 1), (-1, 0), (-4, 3)
Using the distance formula, we find:
AB = √((-1+5)^2 + (0-1)^2) = √(16+1) = √17
BC = √((-4+1)^2 + (3-0)^2) = √(9+9) = √18
AC = √((-4+5)^2 + (3-1)^2) = √(1+4) = √5
The side lengths of this new triangle are congruent to those of triangle ABC. Let's check the angles:
Angle ABC = Angle (-5, 1) = Angle (-4, 3)
These angles are not explicitly given, so we cannot determine their congruence based on the coordinates.
C. (0, 0), (-3, 3), (1, 4)
Using the distance formula, we find:
AB = √((-3)^2 + (3)^2) = √(9+9) = √18
BC = √((1-(-3))^2 + (4-3)^2) = √(16+1) = √17
AC = √((-3-(-1))^2 + (3-4)^2) = √(4+1) = √5
The side lengths of this new triangle are congruent to those of triangle ABC. Let's check the angles:
Angle ABC = Angle (0, 0) = Angle (-3, 3)
These angles are not explicitly given, so we cannot determine their congruence based on the coordinates.
D. (5, 3), (1, 2), (-2, -1)
Using the distance formula, we find:
AB = √((1-5)^2 + (2-3)^2) = √(16+1) = √17
BC = √((-2-1)^2 + (-1-2)^2) = √(9+9) = √18
AC = √((-2-5)^2 + (-1-3)^2) = √(49+16) = √65
Since the side lengths of the new triangle are not congruent to those of triangle ABC, this set of coordinates does not form a congruent triangle.
Therefore, the correct answer is: C. (0, 0), (-3, 3), (1, 4)
To prove that triangle GHI is congruent to triangle JKL, we need to show that all corresponding sides and angles are congruent.
Given:
1. Line GH is congruent to line JK
2. Line HE is congruent to line KL
3. Angle 1 is congruent to angle L
To prove the triangles congruent, we can use the SAS (Side-Angle-Side) congruence criterion.
Here's how we can prove it using SAS:
1. Side GH is congruent to side JK (given).
2. Side HI is congruent to side KL because each is equal to the sum of the corresponding segments: HE + EL = HE + HI = HI + KL (segment addition postulate).
3. Angle GHI is congruent to angle JKL (given).
Therefore, we have shown that triangle GHI is congruent to triangle JKL using the SAS criterion. The correct answer is option A.
Regarding the question about congruent triangles with coordinates:
Triangle ABC has coordinates A(1, 4), B(5, 5), C(2, 2).
To find a set of coordinates that represents a congruent triangle, we need the new triangle to have the same side lengths and angles as triangle ABC.
Let's consider the given options:
A. (3, -1), (4, 5), (5, 7)
B. (-5, 1), (-1, 0), (-4, 3)
C. (0, 0), (-3, 3), (1, 4)
D. (5, 3), (1, 2), (-2, -1)
To check if any of these options represent a congruent triangle, we need to calculate the side lengths and angles of each option and compare them to triangle ABC.
Option A: The side lengths and angles of triangle formed by (3, -1), (4, 5), (5, 7) are not congruent to triangle ABC.
Option B: The side lengths and angles of triangle formed by (-5, 1), (-1, 0), (-4, 3) are not congruent to triangle ABC.
Option C: The side lengths and angles of the triangle formed by (0, 0), (-3, 3), (1, 4) are congruent to triangle ABC. Therefore, this is the correct answer.
Option D: The side lengths and angles of triangle formed by (5, 3), (1, 2), (-2, -1) are not congruent to triangle ABC.
Hence, the triangle with coordinates (0, 0), (-3, 3), (1, 4) is congruent to triangle ABC. The correct answer is option C.