For \(\triangle ABC\) construct line \(\ell\) such that \(\ell \parallel \overline{AC}\) and \(B\) is on \(\ell.\) What is the relationship between the three angles at \(B\) (smaller than a straight angle) to the angles of the triangle?
Theorem3.2.2.Triangle Angle Sum.
The angle sum of the interior angles of a triangle is \(\pi.\)
Corollary3.2.3.Euclidean Exterior Angle.
The measure of an exterior angle of a triangle is equal to the sum of the two opposing, interior angles.
Definition3.2.4.Parallelogram.
A quadrilateral is a parallelogram if and only if both opposing pairs of sides are parallel.
Theorem3.2.5.
Opposite sides of a parallelogram are congruent.
Theorem3.2.6.
If a transversal intersects three parallel lines in such a way as to divide itself into congruent segments, then any transversal of these parallel lines is also divided into congruent segments.
Subsection3.2.2Explore Similarity Theorems
Definition3.2.7.Altitude.
A line segment is an altitude if it connects a vertex of a triangle to the foot of the perpendicular on the opposite side.
Checkpoint3.2.8.
Construct a triangle and enough parallel lines to divide one side of the triangle into four equal parts. Into how many parts do these lines divide the other sides?
Definition3.2.9.Triangle Area.
The area of a triangle is equal to one half of the product of one side times the length of the altitude from the opposing vertex to that side.
Checkpoint3.2.10.
Construct \(\triangle ABC\text{.}\) Construct \(\overline{DE}\) such that \(B-D-A,\)\(B-E-C\text{,}\) and \(\overline{DE} \parallel \overline{AC}.\)
Construct \(\overline{AE}\) and \(\overline{EF}\) such that \(F\) is the foot of the perpendicular from \(E.\) Reduce the ratio of the areas of \(\triangle DEB\) and \(\triangle AED.\)
Construct \(\overline{DC}\) and \(\overline{DG}\) such that \(G\) is the foot of the perpendicular from \(D.\) Reduce the ratio of the areas of \(\triangle DEB\) and \(\triangle CDE.\)
Prove area of \(\triangle AED\) is equal to the area of \(\triangle CDE.\)
Subsection3.2.3Similarity Theorems
Definition3.2.11.Similar Triangles.
Two triangles are similar if and only if corresponding angles are congruent and the ratio of corresponding sides is constant.
Lemma3.2.12.
A line parallel to one side of a triangle and intersecting the other two sides divides those sides proportionally.
Corollary3.2.13.
A line parallel to one side of a triangle divides the triangle into two (whole and part), similar triangles.
Theorem3.2.14.Angle-Angle-Angle (AAA) Similarity.
Two triangle are similar if and only if they have three congruent angles.
Subsection3.2.4Extending Similarity
Checkpoint3.2.15.
Construct a definition for similar quadrilaterals. Construct examples to show that your definition works.
Checkpoint3.2.16.
Explain why similarity is not defined simply as "all angles are congruent."
