**2 Column Proof Help??**

[careness]

I must complete the reason column where it says [1], [2], and [3].
If you could help please do. Thanks.

 

[arthur ohlsten]

RS=RU given
angle SRT = angle URT given
RT is common to both triangles

triangle SRT = triangle URT.............. side angle side

Arthur

 

[fasteddie65]

I respectfully disagree with the answers given in the previous post.
The reason for [1] is "Perpendicular lines form right angles." This is usually a theorem. It is NOT the definition of perpendicular lines.
The reason for [2] is "If two sides of a triangle are congruent, then the angles opposite those sides are congruent." This is also a theorem, sometimes called the Isosceles Triangle Theorem.
The reason for [3] is "Reflexive Property of Congurence."

 

[careness]

I respectfully disagree with the answers given in the previous post.
The reason for [1] is "Perpendicular lines form right angles." This is usually a theorem. It is NOT the definition of perpendicular lines.
The reason for [2] is "If two sides of a triangle are congruent, then the angles opposite those sides are congruent." This is also a theorem, sometimes called the Isosceles Triangle Theorem.
The reason for [3] is "Reflexive Property of Congurence."

thanks, your answer made sense.

 

[soroban]

Hello, careness!

Who wrote this problem?
It is very badly written and its purpose escapes me.

$\displaystyle \text{Given: }\;\begin{array}{cc}(1) & \overline{RT} \perp \overline{SU} \\ (2) & \angle SRT = \angle U\!RT \\ (3) & \overline{RS} = \overline{RU} \\ (4) & T\text{ is midpt of }\overline{SU} \end{array}$

$\displaystyle \text{Prove: }\:\Delta RTS \,\cong\,\Delta RTU$

              R
              *
            * | *
          *   |   *
        *     |     *
    S *   *   *   *   * U
              T

The problem is severely overstated.
. . In an isosceles triangle, the bisector of the vertex angle
. . is always perpendicular to the base and always bisects the base.

So there are several possible proofs **
. . and they picked one which asks rather pointless questions.

Line 2 asks: Do you know how to say "Perpendicular lines form right angles" ?

Line 5 asks: Do you know that "Base angles of an isosceles triangle are equal" ?

Line 9 asks: How do you say "A thing is equal to itself" ?

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

**

$\displaystyle \text{Another proof:}$

.$\displaystyle \begin{array}{cccc} & \text{Statement} && \text{Reason} \\ \hline (1) & ST = TU & & \text{def. of midpoint} \\ (2) & RS = RU & & \text{Given} \\ (3) & RT = RT && \text{Line 9} \\ (4) & \Delta RTS \cong \Delta RTU && \text{s.s.s} \end{array}$



$\displaystyle \text{And another:}$

. . $\displaystyle \begin{array}{cccc} & \text{Statement} && \text{Reason} \\ \hline (1) & RS = RU && \text{Given} \\ (2) & \angle SRT = \angle URT && \text{Given} \\ (3) & RT = RT && \text{Line 9} \\ (4) & \Delta RTS \cong \Delta RTU && \text{s.a.s} \end{array}$



$\displaystyle \text{And yet another:}$

. . $\displaystyle \begin{array}{cccc} & \text{Statement} && \text{Reason} \\ \hline (1) & \angle SRT = \angle URT && \text{Given} \\ (2) & RS = RU & & \text{Given} \\ (3) & \angle S = \angle U & & \text{Line 5} \\ (4) & \Delta RTS \cong \Delta RTU && a.s.a \end{array}$