Geometry, Teacher\'s Edition

d. Equations of lines and circles, including slopes of perpendicular lines

e. The proportionality of the sides of similar triangles

f. Polygons: the sum of interior angles and regular polygons

All the Radii of a Circle Are Congruent –It may seem obvious, but frequently students forget to use the fact that
all the radii of a circle are congruent. This follows directly from the definition of a circle. Remind students to use
this fact when setting up equations and assigning variables to different radii in the same circle.

CongruentTangents –In this section the Tangent Segment Theorems is proved and applied. Remind student that
this is only true for tangents and does not extend to secants. Sometimes student will see a secant enter a circle and
think the distance from the exterior point to where the secant intersects the circle is the same as a tangent or another
secant from that same point.

Hidden Tangent Segments –Sometimes it is difficult for students to recognize tangent segments because they are
imbedded in a more complex figure, or the tangent segment is extended in some way. A common situation where
this occurs is when there is an inscribed circle. Tell the students to be on the lookout for tangent segments. They
should look at segments individually and as part of the whole. Sometimes it is helpful to use a small sticky note to
cover parts of the figure so they do not distract from the area of focus.

Common Tangents and Tangent Circles

Using Trigonometry to Find Side Measurers in Right Triangles –Using the definitions of sine, cosine, or tangent
to find the measures of sides in a right triangle is a common application of trigonometry that is put to use in this
section. Students will need a bit of practice and perhaps a step-by-step process when learning this skill. With some
experience though, this will become an easy, enjoyable task.

Step-by-Step Process:

a. Highlight the side of the right triangle that’s measure is to be found. Place a variable, sayx, by that side.

b. Chose one of the acute angles of the right triangle whose measure is known to work from. Highlight this angle

in another color.

c. Chose another sides of the triangle whose measure is known. Highlight that side in the same color as the other

side.

d. Decide what relationships (opposite leg, adjacent leg, or hypotenuse) the highlighted sides have to the high-

lighted angle.

e. Decide which of the three trigonometric ratios utilize those side relationships.

f. Write out the definition of that trigonometric ratio.

g. Substitute in the highlighted values.

h. Solve the equation by either multiplying or dividing. It is best to not round the decimal approximation of the

trigonometric ratio taken from the calculator. Round after the multiplication or division has taken place.

Key Exercises:

1. 4 ABCis a right triangle with the right angle at vertexC.

m^6 A= 52 ◦andAC=10 cm. FindBC.

Answer:

tan 52◦=

BC

10

BC≈ 12 .8 cm

2. 4 DEFis a right triangle with the right angle at vertexF.

Chapter 2. Geometry TE - Common Errors