By CK-12 Foundation
CK-12’s Geometry - moment version is a transparent presentation of the necessities of geometry for the highschool pupil. themes comprise: Proofs, Triangles, Quadrilaterals, Similarity, Perimeter & sector, quantity, and modifications. quantity 1 comprises the 1st 6 chapters: fundamentals of Geometry, Reasoning and evidence, Parallel and Perpendicular traces, Triangles and Congruence, Relationships with Triangles, and Polygons and Quadrilaterals.
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AB AC (⇐) Assume = = BC . Since ∠ABC ∼ = ∠AB C and ∠ACB ∼ = DE DF EF AB AC ∠AC B , ABC ∼ AB C by AA. By the above argument, AB = AC = AB AC BC BC . This together with = = and AB = DE, we conclude B C DE DF EF that AC = DE and B C = EF . Thus AB C ∼ = DEF by SSS, and ABC ∼ DEF by substitution. We conclude this section with the Side-Angle-Side criterion for similar triangles. This result specializes to the Side-Angle-Side Postulate for congruent DE triangles when AB AC = DF = 1. Theorem 66 (SAS for Similar Triangles) If ABC and DEF are triDE angles such that ∠CAB ∼ ABC ∼ DEF .
The radius of PQ is the distance P Q. If R is a point on PQ such that R − P − Q, then RQ is called a diameter of PQ . 38 Transformational Plane Geometry If D is any point on AB distinct from B and C, the following proposition asserts that ∠BDC is a right angle. Thus an inscribed triangle is a right triangle whenever one of its sides is a diameter. Proposition 86 If B, C and D are distinct points on a circle centered at A, then m∠BAC ≡ 2m∠BDC. Proof. When ABDC is a simple quadrilateral, the points B and C lie on ←→ opposite sides of AD.
Therefore µ(∠ACD) > µ(∠ABC) and the proof is complete. 2. Proof of the Exterior Angle Theorem. Given a point P and a line l, our next theorem asserts that there exists a unique line through P perpendicular to line l. Note that Proposition 29 has already established this fact when P is on l. While the proof of Proposition 29 is a simple application of the Angle Construction Postulate, Theorem 39 establishes a fact essential to define a reflection in transformational plane geometry, namely, if l is a line and P is a point off l, the image of P under the reflection in l is the unique point Q such that l is the perpendicular bisector of P Q.