Practice Makes Perfect Geometry (Practice Makes Perfect (McGraw-Hill))

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Transversals and angles A transversal is a line that intersects two or more lines at different points. When a transversal intersects any lines, groups of angles are formed.

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When the lines cut by the transversal are parallel lines, you can make some statements about the relationships among these angles. As you can see in Figure 3.

In the figure, line t is the transversal. Corresponding angles Two angles, one from each cluster, that are in the same position within their clusters are corresponding angles. If parallel lines are cut by a transversal, corresponding angles are congruent. Alternate interior angles The word alternate indicates that the angles are on different sides of the transversal, and the word interior means that they are between the two lines.

There are two pairs of alternate interior angles in Figure 3. If parallel lines are cut by a transversal, alternate interior angles are congruent. Alternate exterior angles Alternate exterior angles are also positioned on different sides of the transversal, but they are called exterior because they are outside the two lines. If parallel lines are cut by a transversal, alternate exterior angles are congruent. If parallel lines are cut by a transversal, interior angles on the same side of the transversal are supplementary.

Note the difference in that statement. The other pairs of angles were congruent, but interior angles on the same side are supplementary.

Practice Makes Perfect Geometry (Practice Makes Perfect (McGraw-Hill))

Parallel and perpendicular lines If two lines are parallel to the same line, then they are parallel to each other. If two coplanar lines are perpendicular to the same line, they are parallel to each other. If a line is parallel to one of two parallel lines, it is parallel to the other. If a line is perpendicular to one of two parallel lines and lies in the same plane as the parallel lines, it is perpendicular to the other. Line l is parallel to line m. Line m is parallel to line p. Line l is parallel to line p. Line l is parallel to line k.

Line k is perpendicular to line q. Line l is parallel to line q. Line a is perpendicular to line b. Line a is perpendicular to line c. Line b is perpendicular to line c.

Line d is perpendicular to line f. Line d is parallel to line g.


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Line f is to line g. Line l is perpendicular to line m. Line l is perpendicular to line p. Line l is perpendicular to line k. Line l is to line q. Line a is parallel to line b. Line b is 8. Line d is parallel to line f. Line f is parallel to line c. Line m is perpendicular to line p.

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Line l is to line p. A pair of alternate interior angles is congruent. A pair of alternate exterior angles is congruent. A pair of interior angles on the same side of the transversal is supplementary. Both lines are parallel to the same line. Both lines are perpendicular to the same line, and the lines are coplanar. If so, name the parallel lines and give a reason. When you begin to work with more complicated figures, however, you need a new definition of congruent.

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In simple language, however, two figures are congruent if they are exactly the same size and shape. Congruent polygons A polygon is a closed figure formed from line segments that intersect only at their endpoints. The line segments are the sides of the polygon, and the points where the sides meet are the vertices of the polygon.

The polygon has an interior angle at each vertex.

Two polygons are congruent if it is possible to match up their vertices so that corresponding sides are congruent and corresponding angles are congruent. It also tells, by the order in which the vertices are named, which sides and angles correspond. Triangles are classified either by the size of their angles or by the number of congruent sides they have. An acute triangle has three acute angles. A triangle with three congruent sides is an equilateral triangle, and one with two congruent sides is isosceles. If all sides are different lengths, the triangle is scalene.

If a triangle is isosceles, it has two congruent angles, and those angles sit opposite the congruent sides. In a scalene triangle, each angle is a different size. You probably learned all those facts before you got to a geometry course, but some of them, like the fact that base angles of an isosceles triangle are congruent, are proved by using congruent triangles.

Proving triangles congruent When triangles are congruent, all the corresponding angles are congruent and all the corresponding sides are congruent. Luckily, it is not necessary to know about every pair of corresponding sides and every pair of corresponding angles to be certain that a pair of triangles is congruent.

There are postulates and theorems that tell us the minimum information that will guarantee the triangles are congruent. SSS: If three sides of one triangle are congruent to the corresponding sides of another triangle, then the triangles are congruent. SAS: If two sides and the angle included between them in one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. Call it BD. This creates two triangles.

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ASA: If two angles and the side included between them in one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. AAS: If two angles and a side not included between them in one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. AAS is actually a theorem, rather than a postulate. If so, write a congruence statement and state the postulate or theorem that guarantees the congruence.

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This is usually stated as corresponding parts of congruent triangles are congruent, abbreviated as CPCTC. If two sides of a triangle are congruent, the angles opposite those sides are congruent. You can prove that the converse of this is also true: If two angles of a triangle are congruent, the sides opposite those angles are congruent. Write the congruence statement and the postulate or theorem that guarantees congruence. Then list the remaining pairs of congruent corresponding parts.

If so, write the congruence statement and the postulate or theorem that guarantees congruence. In those cases, it is often possible to first prove a different pair of triangles congruent and then use CPCTC to obtain the information you need to prove that the desired triangles are congruent. A V E F W B G D C Determine a which triangles can easily be proved congruent and b which congruent corresponding parts would be necessary to prove the specified triangles congruent.

Given DC AB. Given C is the midpoint of AE. Different teachers will specify different rules about the amount of detail they expect and whether you can refer to theorems by names or you should write out the entire statement of the theorem. Given SV and RW bisect each other. Inequalities in one triangle The first group of inequalities deals with the sides of a single triangle and the relationship between the sides and angles in a single triangle.

The triangle inequality In any triangle, the sum of the lengths of two sides is greater than the length of the third side. The triangle inequality means that the length of any side of a triangle is less than the sum of the other two sides and greater than their difference.

The smallest angle of any triangle is opposite the shortest side. Focus on one triangle and put the sides of that triangle in order, and then move to the other triangle and do the same. If the shared side is the largest side of one triangle and the smallest side of the other, you can link the two inequalities and order all five segments. At other times, however, the differences between triangles are important.

Hinge theorem If two sides of one triangle are congruent to the corresponding sides of another triangle, but the included angles are not congruent, then the triangle with the larger included angle has the longer third side. LN QP 6.