# ABC is a right triangle whose hypotenuse is BC, and AD is the altitude from A on BC. Find: 1. triangle ABC: triangle DAC 2. triangle ABC: triangle DBA You know triangle ABC is a right triangle, and that angle A is the right angle (since it is opposite the hypotenuse BC). You also know that angle C is 60 degrees. So ABC...

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You know triangle ABC is a right triangle, and that angle A is the right angle (since it is opposite the hypotenuse BC). You also know that angle C is 60 degrees. So ABC is a 30-60-90 triangle (the sum of the angles has to be 180, and 180 - (90 + 60) = 30).

Since AD is the altitude from A to BC, you know by definition that the angles on either side of the altitude are right angles.

Now, let's look at the smaller triangles.

ADC is a right angle triangle (angle D of that triangle is one of the right angles made by the altitude AD). You also know angle C is 60 degrees. Again, then, this is a 30-60-90 triangle.

You can do the same analysis for small triangle DBA. D is a right angle, B is a 30 degree angle (see paragraph 2 above), and so again we have a 30-60-90 triangle.

One of the rules of similarity is that if 2 angles of a triangle are equal to 2 angles of another triangle, the triangles are similar.

ABC is similar to ADC because the right angles and angle C of both are equal. ABC is similar to DAB because the right angles and angle B of both are equal.

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