Download Geometry CCSS Common Task: Proving Thales` Theorem Name

Geometry CCSS Common Task:
Proving Thales’ Theorem
Cluster: G.C.A - Understand and apply theorems about circles
Name: ________________________________________ Date: ___________________
Thales’ theorem states the following: If 𝑨, 𝑩, and 𝑪 are three distinct points on a circle, and ̅̅̅̅
𝑨𝑩 is a
diameter of the circle, then ∠𝑨𝑪𝑩 is right.
Use the challenge below to Prove Thales’ theorem.
a.
Draw circle 𝑃 with distinct points 𝐴, 𝐵, and 𝐶 on the circle and diameter ̅̅̅̅
𝐴𝐵 . Prove that ∠𝐴𝐶𝐵 is a right angle.
b.
̅̅̅̅ ). What types of triangles are △ 𝐴𝑃𝐶 and △ 𝐵𝑃𝐶? How do you know?
Draw a third radius (𝑃𝐶
c.
Using the diagram that you just created, develop a strategy to prove Thales’ theorem.
d.
Label the base angles of △ 𝐴𝑃𝐶 as 𝑏° and the base angles of △ 𝐵𝑃𝐶 as 𝑎°. Express the measure of ∠𝐴𝐶𝐵 in terms of 𝑎°
and 𝑏°.
e.
How can the previous conclusion be used to prove that ∠𝐴𝐶𝐵 is a right angle?
Adapted from the mid-module assessment: https://www.engageny.org/resource/geometry-module-5