Question

1. in the patty paper activity, what kind of triangle did you create? how do you know?
2. what is the name of the angle of an isosceles triangle formed by rays containing the two congruent sides of the triangle?
3. use your patty paper exploration to help you decide if each statement is true or false for any triangle abc.
4. list three conjectures about isosceles triangles.

Explanation:

Step1: Analyze the triangle type in Patty - Paper activity

In the Patty - Paper activity, an isosceles triangle is created. We know this because when we fold the paper (a common Patty - Paper technique), we can show that two sides of the triangle are congruent.

Step2: Name the angle in an isosceles triangle

The angle of an isosceles triangle formed by rays containing the two congruent sides is called the vertex angle.

Step3: Evaluate the statements for triangle \(ABC\) (assuming \(AB = BC\))

• For \(AC = AB=BC\):
• In a general isosceles triangle \(ABC\) (where \(AB = BC\)), \(AC\) is the base. So \(AC

eq AB = BC\) (unless it is equilateral). This statement is false.

• For \(\triangle ABD\cong\triangle CBD\):
• If \(AB = BC\) and \(BD\) is the perpendicular bisector (by the properties of isosceles triangles, when we fold along the altitude from the vertex \(B\) to the base \(AC\), \(\triangle ABD\) and \(\triangle CBD\) are congruent by SSS (since \(AB = BC\), \(AD = DC\), and \(BD\) is common). This statement is true.
• For \(m\angle BAC=m\angle BCA\):
• In an isosceles triangle \(ABC\) with \(AB = BC\), by the base - angle theorem, the angles opposite the congruent sides (\(\angle BAC\) and \(\angle BCA\)) are congruent. So \(m\angle BAC = m\angle BCA\). This statement is true.
• For \(\overline{BD}\) is \(\perp\) bisector of \(\overline{AC}\):
• In an isosceles triangle \(ABC\) with \(AB = BC\), the altitude from the vertex \(B\) to the base \(AC\) is also the perpendicular bisector of \(AC\). This statement is true.
• For \(\overline{BD}\) bisects \(\angle ABC\):
• In an isosceles triangle \(ABC\) with \(AB = BC\), the altitude from the vertex \(B\) to the base \(AC\) is also the angle - bisector of \(\angle ABC\). This statement is true.
• For \(AD=\frac{1}{2}(AB)\):
• In a general isosceles triangle \(ABC\) (where \(AB = BC\)), \(AD=\frac{1}{2}AC\) (since \(BD\) is the perpendicular bisector of \(AC\)). There is no reason for \(AD=\frac{1}{2}(AB)\) in a non - special isosceles triangle. This statement is false.

Step4: List three conjectures about isosceles triangles

• Conjecture 1: In an isosceles triangle, the base angles are congruent.
• Conjecture 2: In an isosceles triangle, the altitude from the vertex to the base is also the median and the angle - bisector.
• Conjecture 3: If two sides of a triangle are congruent, then the triangle is isosceles.

Answer:

1. An isosceles triangle. We know this because we can show two sides are congruent (by Patty - Paper folding).
2. Vertex angle.

3.

• \(AC = AB = BC\): False.
• \(\triangle ABD\cong\triangle CBD\): True.
• \(m\angle BAC=m\angle BCA\): True.
• \(\overline{BD}\) is \(\perp\) bisector of \(\overline{AC}\): True.
• \(\overline{BD}\) bisects \(\angle ABC\): True.
• \(AD=\frac{1}{2}(AB)\): False.

4.

• Conjecture 1: In an isosceles triangle, the base angles are congruent.
• Conjecture 2: In an isosceles triangle, the altitude from the vertex to the base is also the median and the angle - bisector.
• Conjecture 3: If two sides of a triangle are congruent, then the triangle is isosceles.