Euclidean Geometry §5.3 Similar Triangles §5.4 the Pythagorean Theorem §5.5 Trigonometry

Home , Euclidean geometry, Pythagorean theorem, Similarity (geometry)

Chapter 5: Euclidean geometry §5.3 Similar Triangles §5.4 The Pythagorean Theorem §5.5 Trigonometry

MTH 411/511

Foundations of Geometry

MTH 411/511 (Geometry) Euclidean geometry Fall 2020 It’s good to have goals

Goals for today: • Prove the Fundamental Theorem on Similar Triangles. • Consider its consequences, including the SAS Similarity Criterion. • Use the Fundamental Theorem to prove the Pythagorean Theorem. • Discuss trigonometry.

MTH 411/511 (Geometry) Euclidean geometry Fall 2020 Fundamental Theorem on Similar Triangles

Recall last time we proved the following theorem.

The Parallel Projection Theorem (Theorem 5.2.1) Let `, m and n be distinct parallel lines. Let t be transversal that cuts these lines at points A, B, and C, respectively, and let t0 be a transversal that cuts these lines at points A0, B0, and C 0, respectively. Assume A ∗ B ∗ C, then AB A0B0 = . AC A0C 0

With this result in hand, we are now ready to prove a major theorem in Euclidean geometry.

Fundamental Theorem on Similar Triangles (Theorem 5.3.1) If 4ABC and 4DEF are two triangles such that 4ABC ∼ 4DEF , then AB DE = . AC DF

MTH 411/511 (Geometry) Euclidean geometry Fall 2020 Fundamental Theorem on Similar Triangles

Fundamental Theorem on Similar Triangles (Theorem 5.3.1) If 4ABC and 4DEF are two triangles such that 4ABC ∼ 4DEF , then AB DE = . AC DF

Proof. Let 4ABC and 4DEF be two triangles such that 4ABC ∼ 4DEF . We claim AB/AC = DE/DF . There are three cases: either AB = DE, AB > DE, or AB < DE (trichotomy). In the ﬁrst case, the result is obvious. The second two cases are similar, so we will prove the case AB > DE. −→ Choose a point B0 on AB such that AB0 = DE (PCP). Let m be the line through B0 ←→ such that m is parallel to ` = BC (EPP) and let C 0 be the point at which m intersects 0 0 ∼ ∼ AC (Pasch’s Axiom). Then ∠AB C = ∠ABC = ∠DEF (Converse to AIAT) so 4AB0C 0 =∼ 4DEF (ASA). Let n be the line through A that is parallel to ` and m (EPP and ToP). Thus AB0/AB = AC 0/AC (Parallel Projection Theorem) and DE/AB0 = DF /AC 0 (deﬁnition of congruent triangles). Hence, DE/DF = AB/AC (algebra).

MTH 411/511 (Geometry) Euclidean geometry Fall 2020 Fundamental Theorem on Similar Triangles

The following is an alternate way to view the previous result.

(Corollary 5.3.2) If 4ABC and 4DEF are two triangles such that 4ABC ∼ 4DEF , then there is a positive number r such that

DE = r · AB, DF = r · AC, and EF = r · BC.

We call r the common ratio of the sides of the similar triangles.

MTH 411/511 (Geometry) Euclidean geometry Fall 2020 SAS Similarity Criterion

Just as with congruent triangles, there are more eﬃcient ways to check for similarity than by using the deﬁnition. The following could be taken as an axiom because it is equivalent to the Euclidean Parallel Postulate.

SAS Similarity Criterion (Theorem 5.3.3) ∼ If 4ABC and 4DEF are two triangles such that ∠CAB = ∠FDE and AB/AC = DE/DF , then 4ABC ∼ 4DEF .

Converse to the Similar Triangles Theorem (Theorem 5.3.4) If 4ABC and 4DEF are two triangles such that AB/DE = AC/DF = BC/EF , then 4ABC ∼ 4DEF .

MTH 411/511 (Geometry) Euclidean geometry Fall 2020 The Pythagorean Theorem

We introduce some notation for triangles to simplify the exposition in this section.

Let 4ABC be a triangle. Denote ∠CAB by ∠A, ∠ABC by ∠B, and ∠ACB by ∠C. The corresponding lowercase letter is used to denote the length of the opposite side, so a = BC, b = AC, and c = AB. If 4ABC is a right triangle, then the right angle is always located at C.

c a

A C b

Pythagorean Theorem (Theorem 5.4.1) If 4ABC is a right triangle with right angle at vertex C, then a2 + b2 = c2.

MTH 411/511 (Geometry) Euclidean geometry Fall 2020 The Pythagorean Theorem

Pythagorean Theorem (Theorem 5.4.1) If 4ABC is a right triangle with right angle at vertex C, then a2 + b2 = c2.

Proof. ←→ Drop a perpendicular from C to AB and call the foot of that perpendicular D. Then D ◦ is in the interior of AB (Lemma 4.8.6). Now µ(∠A) + µ(∠B) = 90 and ◦ ∼ µ(∠A) + µ(∠ACD) = 90 (Angle Sum Theorem). Thus, ∠B = ∠ACD and similarly ∼ ∠A = ∠DCB. Hence, there are similar triangles 4ABC ∼ 4CBD ∼ 4ACD.

Let x = AD, y = BD, and h = CD. Then x/b = b/c and y/a = a/c (Fund. Thm. of Similar Triangles). Thus, b2 = cx and a2 = cy, so a2 + b2 = c(x + y) (algebra). Since x + y = c, the proof is complete.

MTH 411/511 (Geometry) Euclidean geometry Fall 2020 The Pythagorean Theorem

In the right triangle 4ABC, the segment CD is an altitude of the triangle and its length, h = CD, is called the height of the triangle. The segment AD is called the projection of AC onto AB. Similarly, BD is the projection of BC onto AB. C

b a

A B x D y

Deﬁnition 1 √ The geometric mean of two positive numbers x and y is deﬁned to be xy.

Using this terminology, we can rephrase the Pythagorean Theorem.

(Theorem 5.4.3) The height of a right triangle is the geometric mean of the lengths of the projections of the legs.

MTH 411/511 (Geometry) Euclidean geometry Fall 2020 The Pythagorean Theorem

(Theorem 5.4.4) The length of one leg of a right triangles the geometric mean of the length of the hypothenuse and the length of the projection of that leg onto the hypotenuse.

Converse to the Pythagorean Theorem (Theorem 5.4.4)

2 2 2 If 4ABC is a triangle such that a + b = c , then ∠BCA is a right angle.

MTH 411/511 (Geometry) Euclidean geometry Fall 2020 Trigonometry

The Fundamental Theorem of Similar Triangles and the Pythagorean Theorem form the foundation for studying trigonometry. Definition 2 Let θ be an acute angle with vertex A. Then θ consists of two rays with a common endpoint A. Choose a point B on one of the rays and drop a perpendicular to the other ray. Call the foot of that perpendicular C. Define the sine and cosine functions by BC AC sin θ = and cos θ = . AB AB If θ is an obtuse angle, let θ0 denote its supplement. Define

sin θ = sin θ0 and cos θ = − cos θ0.

If θ has measure 0◦, deﬁne

sin θ = 0 and cos θ = 1.

If θ is a right angle, deﬁne

sin θ = 1 and cos θ = 0.

MTH 411/511 (Geometry) Euclidean geometry Fall 2020 Trigonometry

The trigonometric functions have domain the set of angles, but by the Fundamental Theorem of Similar Triangles, they only depend on the angle measure. Hence, we may regard then as functions on the interval [0, 180). The range (image) of sine is [0, 1] and the range (image) of cosine is (−1, 1].

The following theorem follows almost directly from the Pythagorean Theorem.

Pythagorean Identity (Theorem 5.5.2) For any angle θ, sin2 θ + cos2 θ = 1.

MTH 411/511 (Geometry) Euclidean geometry Fall 2020 Trigonometry

The next pair of theorems hold for any triangle. The proofs of these statements is part of your next homework.

Law of Sines (Theorem 5.5.3) If 4ABC is any triangle, then sin A sin B sin C = = . a b c

Law of Cosines (Theorem 5.5.4) If 4ABC is any triangle, then

c2 = a2 + b2 − 2ab cos C.

MTH 411/511 (Geometry) Euclidean geometry Fall 2020 Next time

Before next class: Read the Chapter 6 introduction and Section 6.1.

In the next lecture we will: • Introduce hyperbolic geometry. • Review basic theorems in hyperbolic geometry and properties of quadrilaterals. • Prove the AAA congruence condition.

MTH 411/511 (Geometry) Euclidean geometry Fall 2020