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A straight line  divides a triangle  into two congruent triangles. Select all the statements that must be true.

1.Triangle ΔΔ has two equal sides.

2.Triangle ΔΔ has two equal angles.

3.The line ℓℓ is perpendicular to a side of Δ.Δ.

4.The line ℓℓ passes through a midpoint of one side of Δ.Δ.

 Oct 1, 2023
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Of the statements provided, only 4. The line ℓℓ passes through a midpoint of one side of Δ.Δ. must be true.

We can see this by drawing a diagram of the situation:

[asy] unitsize(0.8 cm);

pair A, B, C, D;

A = (0,0); B = (10,0); C = intersectionpoint(arc(A,10,0,180), arc(B,10,180,0)); D = (B + C)/2;

draw(A--B--C--cycle); draw(A--D);

label("A", A, SW); label("B", B, SE); label("C", C, NE); label("D", D, S); [/asy]

If line ℓ divides triangle △ABC into two congruent triangles, then it must pass through the midpoint of one of the sides of the triangle. In this case, line ℓ passes through the midpoint of side BC.

The other statements are not necessarily true. For example, triangle △ABC does not necessarily have two equal sides or two equal angles. Also, line ℓ does not necessarily have to be perpendicular to a side of the triangle.

Here are some examples:

If triangle △ABC is an isosceles triangle, then it will have two equal sides, but line ℓ may not pass through the midpoint of one of the sides.

If triangle △ABC is an equilateral triangle, then it will have two equal sides and two equal angles, but line ℓ may not pass through the midpoint of one of the sides.

If triangle △ABC is a right triangle, then line ℓ may be perpendicular to a side of the triangle, but it may not pass through the midpoint of one of the sides.

Therefore, the only statement that must be true is that line ℓ passes through the midpoint of one side of the triangle.

 Oct 1, 2023

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