Construct the triangle that has a side of length t included

LEMMAS HELPING THEOREMS We will use the following theorems to help us prove the theorems found later in this sec- tion. In their role as “helping” theorems, each of the five boxed statements that follow is called a lemma. We will prove the first four lemmas, because their content is geometric. A B C Figure 3.45 B C D A 1 2 Figure 3.46 1 2 3 Figure 3.47 C A B Figure 3.48 If B is between A and C on , then and . The measure of a line segment is greater than the measure of any of its parts. See Figure 3.45. AC 7 BC AC 7 AB AC LEMMA 3.5.1 PROOF By the Segment-Addition Postulate, . According to the Ruler Postulate, meaning BC is positive; it follows that . Similarly, . These relationships follow directly from the definition of . a 7 b AC 7 BC AC 7 AB BC 7 AC = AB + BC If separates into two parts and , then and . The measure of an angle is greater than the measure of any of its parts. See Figure 3.46. m ∠ABC 7 m∠2 m ∠ABC 7 m∠1 ∠2 ∠1 ∠ABC BD LEMMA 3.5.2 PROOF By the Angle-Addition Postulate, . Using the Protractor Postulate, ; it follows that . Similarly, . m ∠ABC 7 m∠2 m ∠ABC 7 m∠1 m ∠2 7 0 m ∠ABC = m∠1 + m∠2 If is an exterior angle of a triangle and and are the nonadjacent interior angles, then and . The measure of an exterior angle of a trian- gle is greater than the measure of either nonadjacent interior angle. See Figure 3.47. m ∠3 7 m∠2 m ∠3 7 m∠1 ∠2 ∠1 ∠3 LEMMA 3.5.3 PROOF Because the measure of an exterior angle of a triangle equals the sum of measures of the two nonadjacent interior angles, . It follows that and . m ∠3 7 m∠2 m ∠3 7 m∠1 m ∠3 = m∠1 + m∠2 In , if is a right angle or an obtuse angle, then and . If a triangle contains a right or an obtuse angle, then the measure of this angle is greater than the measure of either of the remaining angles. See Figure 3.48. m ∠C 7 m∠B m ∠C 7 m∠A ∠C 䉭ABC LEMMA 3.5.4 PROOF In , . With being a right angle or an obtuse angle, ; it follows that . Then each angle and must be acute. Thus, and . The following theorem also a lemma is used in Example 1. Its proof not given depends on the definition of “is greater than,” which is found on the previous page. m ∠C 7 m∠B m ∠C 7 m∠A ∠B ∠A m ∠A + m∠B … 90° m ∠C Ú 90° ∠C m ∠A + m∠B + m∠C = 180° 䉭ABC If and , then . a + c 7 b + d c 7 d a 7 b LEMMA 3.5.5 Addition Property of Inequality