A few more theorems and proofs for your notebook

Isosceles triangle theorem:

If a triangle is isosceles then the angles opposite the congruent sides are congruent.

Proof:

Bisect –ABC with segment BD.

–ABD@–DBC (definition of angle bisector)

AB@CB (definition of isosceles triangle)

BD@BD (same segment)

DABD@DCBD (side-angle-side postulate)

–BAD@–BCD (definition of congruent triangle)

(proven by Jennifer, and Jonathan F. on Nov. 27 Homework)

Another Proof:

Bisect segment AC with segment BD.

AD@DC (definition of segment bisector)

AB@CB (definition of isosceles triangle)

BD@BD (same segment)

DABD@DCBD (side-side-side postulate)

–BAD@–BCD (definition of congruent triangle)

(proven by Mark on Nov. 27 Homework)

Yet Another Proof:

AB@CB (definition of isosceles triangle)

BC@BA (definition of isosceles triangle)

AC@CA (same segment)

DABC@DCBA (side-side-side postulate)

–BAD@–BCD (definition of congruent triangle)

Which proof do you like best? Why?

A few more theorems and proofs for your notebook (page 2)

Converse of Isosceles triangle theorem:

If two angles of a triangle are congruent, then the triangle is isosceles and the sides opposite the congruent angles are congruent.

Proof:

–BAC@–BCA (given)

AC@CA (same segment)

–ACB@–CAB (given)

DABD@DCBD (angle-side-angle postulate)

BA@BC (definition of congruent triangle)

 

(Can you find other proofs?)

 

 

Triangle midpoint theorem:

The segments joining the midpoints of the sides of a triangle form four congruent triangles: one using all three segments
(DDEF), and three using one segment and parts of two sides (DFCD, DBFE, DEDA).

 

(See next page for proof)

A few more theorems and proofs for your notebook (page 3)

Proof of Triangle midpoint theorem:

(This is tricky. The idea is to use another figure that we can prove things about, and then prove it is the same figure. Sometimes we only need to add a line or something. Here we need to add a new figure.)

Draw a figure so that:

DZYX@DABC

XW@WZ

WV is parallel to XY

WU is parallel to ZY

–WXU@–ZWV
(corr. angles)

WX@ZW (given)

–XWU@–WZV
(corr. angles)

DWXU@DZWV (angle-side-angle postulate)

XU@WV
(definition of congruent triangles)

–XUW@–VWU (alternate angles)

UW@WU (same segment)

DXUW@DVWU (side-angle-side postulate)

m–XUW+m–WUV+m–VUY=180°ree; (on a straight line, angle addition postulate)

–XWU@–WUV (alternate angles, or definition of congruent triangles)

m–XUW+m–XWU+m–VUY=180°ree; (substitution of equals)

m–XUW+m–XWU+m–UXW=180°ree; (angle sum of triangle = 180°ree;)

–YUV@–UXW (transitive, subtraction, etc.)

–VYU@–WUX (Corresponding angles)

XW@UV (definition of congruent triangles)

DUXW@DYUV (angle-angle-side theorem)

(continued)

A few more theorems and proofs for your notebook (page 4)

Proof of Triangle midpoint theorem (continued)

XU@UY (definition of congruent triangles)

ZV@VY (definition of congruent triangles)

–ACB@–ZXY (definition of congruent triangles)

AC@ZX (definition of congruent triangles)

DC@WX (both are half of congruent segments)

BC@YX (definition of congruent triangles)

FC@UX (both are half of congruent segments)

DDCF@DWXU (side-angle-side postulate)

DAED@DZVW (same reasoning as DDCF@DWXU)

DEBF@DVYU (same reasoning as DDCF@DWXU)

DE@WV (definition of congruent triangles)

EF@VU (definition of congruent triangles)

FD@UW (definition of congruent triangles)

DDEF@DWVU (side-side-side postulate)

DUXW@DYUV (from above)

DDCF@DWXU (from above)

DEBF@DVYU (from above)

DEBF@DDFC (transitive postulate)

DXUW@DVWU (from above)

DDEF@DWVU (from above)

DDCF@DWXU (from above)

DDCF@DFED (transitive postulate)

DAED@DDFC (by the same reasoning)

QED (an abbreviation for quod erat demonstrandum, which means "What we had to prove is proven.")