Geometry, Teacher\'s Edition

e. A diameter creating a 180 degree semicircle

f. Arcs covering the entire 360 degrees of the circle

g. Central or Inscribed angles

h. Tangents that form right angles

i. Similar triangle with proportional sides

j. Congruent triangles with congruent corresponding parts

Any New Information is Good –If students can not immediately see how to find the measure they are after, advise
them to find any measure they can. This keeps their mind active and working. Frequently, they will be able to use
the new information to find other measures, and will eventually work their way around to the desired answer. This
might not br the most efficient method, but the students’ technique will improve with practice.

Angles of Chords, Secants, and Tangents

Where’s the Vertex? –When determining the relationships between angles and arcs in a circle the location of the
vertex of the angle is the determining factor. There are four possibilities.

a. The vertex of the angle is at the center of the circle, it is a central angle, and the arc and angle have the same

measure.

b. The vertex of the angle is on the circle. The angle could be made by two cords, an inscribed angle, or by a

chord and a tangent. In either situation, the measure of the arc is twice that of the angle.

c. The vertex of the angle is inside the circle, but not at the center. In this case two arcs are necessary, and the

angle measure is the average of the measures of the arcs cut off by the chords that form the vertical angles.

d. The vertex of the angle is outside the circle. Then the two intersected arcs have to be subtracted and the

difference divided by two. Note the similarity to an average.

Students often need help organizing information in this way. It is best to do this with them, as a class activity so that
in the future they will be able to do it for themselves.

Use the Arcs –It is typical to have more than one angle intercepting a specific arc. In this case a measure can be
moved to an arc and then back out to another angle. Another situation students should look for is when a circle is
divided into two arcs. One arc can be represented as 360−(an expression for the other arc). Students sometimes
miss these kinds of moves. It may be beneficial to have students share with the class the different strategies and
patterns they see when working on these exercises.

Additional Exercises:

1. Two tangent segments with a common endpoint intercept a circle dividing it into two arcs, one of which is twice
   as big as the other. What is the measure of the angle formed by the by the two tangents?

Answer:

x+ 2 x= 360 angle measure= ( 240 − 120 )÷ 2 =60 degrees

x= 120

2. Two intersecting chords intercept congruent arcs. What kind of angles do the chords form?

Answer: central angles

2.9. Circles