Angles, similarity & cosine · SAT Math

A diagram alone does not establish parallel lines, a right angle or similarity. Once those conditions are given, the corresponding theorems connect algebra to geometry. Corresponding angles at parallel lines are equal; a straight-line pair is supplementary. Choosing the wrong relation can create a neatly solved but irrelevant equation.

Write the equation before solving

If corresponding angles measure 3x+10 degrees and 70 degrees, equality gives x=20. Treating them as supplementary would instead give a different input. For an exterior triangle angle of 110 degrees with remote interior angles 35 and 2x+5, use 110=35+2x+5, so x=35. The adjacent interior angle is 70 degrees, not the remote unknown.

Perimeter follows lengths

Similar triangles with corresponding sides two and six have length scale three. A smaller perimeter of eleven becomes 33. Perimeter adds side lengths, so it does not use the area factor nine. Check that you are enlarging in the direction requested and that the two given sides actually correspond.

Complementary angles share a ratio

In a right triangle, the two acute angles A and B add to 90 degrees. The leg opposite A is adjacent to B, with the same hypotenuse. Thus sin A=cos B. If sin A=5/13, then cos B=5/13; cos A is 12/13 instead. Complementary angles do not make their sine values add to one.

Transfer and check

For sin A=7/25, identify cos B as 7/25 and cos A as 24/25. Verify the side relation 7²+24²=25². Explain why the complementary-angle identity can answer the first question without computing the remaining leg at all.

WORKED EXAMPLE

Two parallel lines are cut by a transversal. A pair of corresponding angles measures (3x+56)° and 65°. Find x.

Corresponding angles on parallel lines are equal.
So 3x+56=65, which gives 3x=9.
Dividing gives x=3; the resulting angle has the stated measure.

Answer: 3

Compare the other answer choices

Subtract the constant but omit division by 3.

121/3

Add the constant while isolating x.

59/3

Treat corresponding angles as supplementary instead of equal.

Try it, then explain it.

Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.

Practice this skill

Sources & authorship

Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.

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