Math: Angles, Triangles and Similarity

This is Unit Thirteen of ACT Preparation: angles, triangles and similarity. Unit twelve closed the algebra of functions; this unit begins plane geometry. One habit runs through it: name the fact behind every step, because each move in a figure uses exactly one. By the end of this part you will be able to use angle pairs. To find angles through parallel lines. To use the triangle angle sum, the exterior angle and isosceles base angles. To decide whether two triangles are congruent. And to use the properties of quadrilaterals.

Six entries on angles, and on the angles parallel lines make with a transversal.

Measuring angles.

ACT 1201 Angles and Their Measures

An angle is measured in degrees. Acute is less than ninety, right is exactly ninety, obtuse is between ninety and 180, and straight is 180. Clock hands at three o'clock form a right angle; at six, a straight angle. Two angles that together form a line add to 180.

Pairs that add to 90 and 180.

ACT 1202 Complementary and Supplementary Angles

Two angles are complementary when they add to ninety degrees and supplementary when they add to 180; angles forming a straight line are supplementary. The complement of thirty-five degrees is fifty-five, and its supplement 145. Complementary goes with the corner; supplementary with the straight line.

Angles opposite each other.

ACT 1203 Vertical Angles

When two lines cross, the angles opposite each other are vertical angles, and they are equal; neighboring angles are supplementary. Lines crossing at seventy degrees make angles of seventy, 110, seventy and 110. Vertical means opposite across the crossing, not up and down.

Parallel lines and a transversal.

ACT 1204 Alternate Interior Angles

When a transversal crosses two parallel lines, alternate interior angles, on opposite sides of the transversal and between the parallels, are equal. If one is sixty-five degrees, so is the other. The rule needs parallel lines; a Z shape is only a memory aid.

The other angle pairs.

ACT 1205 Corresponding and Same-Side Interior Angles

With parallel lines and a transversal, corresponding angles, in matching positions at each crossing, are equal, and same-side interior angles add to 180. With a sixty-five-degree angle above, its corresponding angle below is sixty-five and its same-side partner 115. Same-side interior is the one pair that adds instead of matching.

Finding angles through parallel lines, step by step.

ACT 1206 Finding Angles Through Parallel Lines

Mark the parallels and the transversal, move from angle to angle one fact at a time, and name the fact at each step. When corresponding angles are 3x plus ten and 5x minus thirty, they are equal, so x is twenty and the angle seventy degrees; the angle beside it on the line is 110. The equation needs the right relationship: if those angles were same-side interior, they would add to 180, and x would be twenty-five.

Five entries on triangles: the angle sum, the exterior angle, and the isosceles triangle.

The angles of a triangle.

ACT 1207 The Triangle Angle Sum

The three angles of a triangle add to 180 degrees, a consequence of the parallel postulate. Angles of fifty and seventy leave sixty for the third, and a right triangle's two acute angles add to ninety. A triangle can have at most one right or obtuse angle.

The exterior angle.

ACT 1208 The Exterior Angle

An exterior angle of a triangle, formed by extending one side, equals the sum of the two interior angles not next to it. Interior angles of forty and seventy-five give an exterior angle of 115. The exterior angle and its neighboring interior angle are supplementary.

Naming triangles.

ACT 1209 Triangle Types

By sides, a triangle is scalene, isosceles with two equal sides, or equilateral with three; by angles, acute, right or obtuse. An equilateral triangle has three sixty-degree angles. The names combine: a right isosceles triangle has angles of ninety, forty-five and forty-five.

The isosceles triangle.

ACT 1210 The Isosceles Triangle

In an isosceles triangle, the base angles, opposite the equal sides, are equal, and two equal angles mean the sides opposite them are equal. A forty-degree top angle leaves base angles of seventy each. The base angles sit opposite the equal sides, not between them.

Angles in a triangle figure, step by step.

ACT 1211 Finding Angles in a Triangle Figure

Record every given angle and marked equal side, apply the facts that create new angles, and chain them, writing a reason for each step. In triangle A B C with A B equal to A C and angle A thirty-six degrees, the base angles B and C are seventy-two each, and the exterior angle at C is thirty-six plus seventy-two, 108; the straight line confirms it. Mark which sides are equal before naming base angles.

Four entries on congruent triangles and the tests that prove them.

Same size, same shape.

ACT 1212 Congruent Triangles

Congruent triangles have all corresponding sides and angles equal, and the congruence statement lists vertices in corresponding order. Triangle A B C congruent to D E F means A matches D and side A B equals D E. Read the letter order, not the picture.

The congruence tests.

ACT 1213 SSS, SAS, ASA and AAS

Two triangles are congruent if three pairs of sides match, side-side-side; if two sides and the angle between them match, side-angle-side; or if two angles and a side in corresponding position match, angle-side-angle or angle-angle-side. In side-angle-side, the angle must sit between the two sides.

Right triangles, and two tests that fail.

ACT 1214 Hypotenuse-Leg, and Two Tests That Fail

Two right triangles are congruent if the hypotenuse and a leg match. Matching angles alone prove only similarity, and side-side-angle does not prove congruence: two triangles with angles of fifty, sixty and seventy can differ in size. Side-side-angle works only in the right-triangle case.

Deciding congruence, step by step.

ACT 1215 Which Triangles Are Congruent

List the matching parts, including shared sides and vertical angles, check whether they fit a valid test, and if so transfer a missing measure. When segments A D and B C cross at E with A E equal to D E and B E equal to C E, the vertical angles at E complete side-angle-side, so A B equals D C. A shared side or a vertical angle is a given the figure does not label; count it before deciding.

Three entries on parallelograms and the special quadrilaterals.

Parallelograms.

ACT 1216 Parallelograms

A parallelogram has both pairs of opposite sides parallel. Its opposite sides and angles are equal, consecutive angles are supplementary, and its diagonals bisect each other. One sixty-five-degree angle gives angles of sixty-five, 115, sixty-five and 115. The diagonals are not equal in general.

The special quadrilaterals.

ACT 1217 Rectangles, Rhombuses, Squares and Trapezoids

A rectangle is a parallelogram with right angles and equal diagonals; a rhombus has four equal sides and perpendicular diagonals; a square is both; and a trapezoid has exactly two parallel sides. Every square is a rectangle and a rhombus, but not every rectangle is a square.

Using quadrilateral properties, step by step.

ACT 1218 Using Quadrilateral Properties

Identify the figure from what is given, list the properties it guarantees, and write an equation from the property that connects the given and the unknown. In a parallelogram with consecutive angles 2x plus fifteen and 3x minus ten, they add to 180, so the angles are eighty-five and ninety-five. A rectangle's diagonals 3x plus four and 5x minus eight are equal, so each is twenty-two. Use only properties the figure is known to have.

Look at what you can now do. You can use complementary, supplementary and vertical angles. You can move through parallel lines one fact at a time, choosing equal or supplementary correctly. You can use the triangle angle sum, the exterior angle, and isosceles base angles. You can prove triangles congruent with a valid test, counting unmarked givens. And you can use what each quadrilateral guarantees, and nothing more. Part two turns to similar triangles, the Pythagorean theorem, and scale.