G10 Math Q2- Week 2_3 Proves Angles Arcs.pptx

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1. Chords, Arcs, and Central Angles

2. Objectives: 1. Identify the relations among chords, arcs, central angles and inscribed angles. 2. Name the chords, arcs, central angles, inscribed angles given a circle. 3. Relate the idea on circles in everyday life situation. 4. Proves theorems related to chords, arcs and central angles

3. Know My Terms and Conditions… Use the figure below to identify and name the following terms related to A. Then, answer the questions that follow.  

4. A ⦁ ⦁ ⦁ ⦁ ⦁ J N L E

5. 1. a radius 2. a diameter 3. a chord 4. a semicircle 5. a minor arc 6. a major arc 7. 2 central angles 8. 2 inscribed angles

6. Terms related to circles Parts of the circle 1. a radius 2. a diameter 3. a chord 4. a semicircle JNE , JLE 5. a minor arc JN , EN , EL , JN 6. a major arc LEN , LJE, ENL, JLN, LNJ 7. 2 central angels 8. 2 inscribed angles Terms related to circles Parts of the circle 1. a radius 2. a diameter 3. a chord 4. a semicircle JNE , JLE 5. a minor arc JN , EN , EL , JN 6. a major arc LEN , LJE, ENL, JLN, LNJ 7. 2 central angels 8. 2 inscribed angles       b      

7. Questions: How did you identify and name the radius, diameter, and chord? How about the semi-circle, minor, and major arc? inscribed angle and central angle?

8. How do you describe the radius, diameter, and chord of a circle? How about the semi-circle, minor arc, and major arc? Inscribed angle and central angle?

9. Write your answers in the table below? Terms Related to Circles Description Radius Diameter Chord The distance from the center to Any point on the circle The distance across the circle through its center A segment whose endpoints lie on the circle

10. Terms Related to Circles Description Semicircle 5. Minor arc 6. Major arc an arc measuring one-half the circumference of a circle an arc of a circle whose measure is greater than that of a semicircle an arc of a circle whose measure is less than that of semicircle

11. Terms Related to Circles Description 7. Central angle Inscribed angle an angle formed by two rays whose vertex is the center of the circle an angle whose vertex is on a circle and whose sides contain chords of the circle

12. 1. In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent. In ⊙ E below, ∠SET≅∠NEO . Since the two central angles are congruent, the minor arcs they intercepts are also congruent. Hence, ST≅NO . If and , then .  

13. 2. In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. In ⊙T below, BA≅CH . Since the two are chords are congruent, then and BA≅CH If ⊙T≅⊙N and BA≅CH≅OE , then BA≅CH≅OE .

14. 3. In a circle, a diameter bisects a chord and an arc with the same endpoints if and only if it is perpendicular to the chord. In ⊙ U below, ES is a diameter and GN is a chord. If ES⊥GN , then GI≅IN and GE≅EN .

15. ACTIVITY 1 Determine the reason behind every statement that proves that in a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent. Choose your answer in the word bank below. Given: ⊙ E ≅ ⊙ I ∠SET≅∠BIG Prove: ST≅BG Statements Reason   In E, . In I, .         Statements Reason           Proof:

16. Word bank From 4, definition of congruent arcs The degree measure of a minor arc is the measure of the central angle which intercepts the arc Given From 2 and 3, substitution From 1, definition of congruent angles

17. ACTIVITY 2 Determine the reason behind every statement that proves that in a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. Choose your answer in the word bank below. Given: ⊙T≅⊙N and AB≅OE Prove: AB≅OE

18. Statements Reason           Statements Reason           Word bank Given Radii of the same circle or of congruent circles are congruent SSS Postulate Corresponding Parts of Congruent Tringles are Congruent (CPCTC) From the previous theorem, “in a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent.”

19. In ⊙A below, m∠LAM =42, m∠HAG =30 , and m∠KAH is a right angle. Find the following measure of an angle or an arc. m∠LAK = m∠JAK = m∠LAJ = m∠JAH = m∠KAM = m∠LK = m∠JK =

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